Hi everyone! This is the first season of Brilliant Sub Junior Calculus Contest. This contest is for beginners or intermediate ones who want to sharpen their skills of problemsolving in overall calculus.
The aim of this contest is to improve the skill of in the computation in all sorts of problem (of basic level) in calculus like integrals (both definite and indefinite), differentiation, limits, ecetra by learning from each other and of course to have fun!
Eligibility: People should fulfill either of the 2 following
17 years or below
Level 4 or below in Calculus
Eligible people here may participate in this contest.
The rules are as follows:
I will start by posting the first problem. If there is a user solves it, then they must post a new one.
You may only post a solution of the problem below the thread of problem and post your proposed problem in a new thread. Put them separately.
Only make substantial comment that will contribute to the discussion.
Make sure you know how to solve your own problem before posting it in case there is no one can answer it within 48 hours, then you must post the solution and you have a right to post another problem.
If the one who solves the last problem does not post his/her own problem after solving it within a day, then the one who has a right to post a problem is the last solver before him/her.
The scope of questions is only computation of basic level problems in calculus.
It is NOT compulsory to post original problems. But make sure it has not been posted on brilliant.
You are also NOT allowed to post a solution using a contour integration or residue method.
Answer shouldn't contain any Special Function.
Please post your solution and your proposed problem in a single new thread.
Format your post is as follows:
1 2 3 4 5 6 7 

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Top NewestProblem 4
\[\int_0^{\pi}\sqrt{1+4\sin^2 \frac x24\sin \frac x2} dx=?\]
(Leave the answer in square roots and \(\pi\)No simplifications required..) – Rishabh Cool · 10 months ago
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First we start with the following inequality:
\(\sin(\dfrac{x}{2})\geq \dfrac{1}{2} \\ \dfrac{x}{2}\geq \arcsin(\dfrac{1}{2}) \\ \dfrac{x}{2}\geq \dfrac{\pi}{6} \\ x\geq \dfrac{\pi}{3} \\ Now, 1+4\sin^2\dfrac{x}{2}4\sin\dfrac{x}{2} = \displaystyle (2\sin\dfrac{x}{2}1)^2 \\ And \displaystyle \sqrt{1+4\sin^2\dfrac{x}{2}4\sin\dfrac{x}{2}} =2\sin\dfrac{x}{2}1 \)
So, the integral is : \( \displaystyle \int_0^\pi 2\sin\dfrac{x}{2}1 dx \) which can be split into two limits  \(0 \ to \ \dfrac{\pi}{3} \ and \ \dfrac{\pi}{3} \ to \ \pi \) to remove the mod sign.
\(\displaystyle \int_0^\dfrac{\pi}{3} (12\sin\dfrac{x}{2}) \ dx +\displaystyle \int_\dfrac{\pi}{3}^\pi (2\sin\dfrac{x}{2}1) \ dx\)
Now substituting the proper limits to this simple integral we get the answer as \(4\sqrt34\dfrac{\pi}{3}\) – Samarth Agarwal · 10 months ago
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PROBLEM 1:
\[\int_a^{b} \frac{1}{x^2+ax+b} \mathrm{d}x\]
Answer in terms of \(a\) and \(b\) only where \(4b>a^2\).
This problem was solved by Nihar Mahajan. – Akshay Yadav · 10 months ago
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The integral can be written as \(\displaystyle\int_{a}^{b} \dfrac{1}{\left(x+\dfrac{a}{2}\right)^2 + b\dfrac{a^2}{4}} \ dx\) . Now substitute \(y=x+\dfrac{a}{2}\) to get \(\displaystyle\int_{3a/2}^{(2b+a)/2} \dfrac{1}{ y^2+ b\dfrac{a^2}{4}} \ dy\). Now using \(\int \dfrac{1}{x^2+a} = \dfrac{\tan^{1}\left(\dfrac{x}{\sqrt{a}}\right)}{\sqrt{a}}+C\) , and substituting limits , the answer IS \(\dfrac{2}{\sqrt{4ba^2}} \tan^{1}\left(\dfrac{2(ba)(4ba^2)}{2a^2+6ab+4b}\right)\) .
Partial credits to Vighnesh Shenoy to give me a start to this problem and Akshay for correcting my answer. – Nihar Mahajan · 10 months ago
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PROBLEM 3:
\[ \displaystyle \int \dfrac{2x1}{x^22x+10} \ dx\]
This question was solved by Rishabh Cool and Akshay Yadav at almost same time. – Samarth Agarwal · 10 months ago
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\[\displaystyle \int \dfrac{2(x1)+1}{x^22x+10} \ dx\]
\[=\displaystyle \int \dfrac{2(x1)}{x^22x+10}+\dfrac{1}{(x1)^2+9} \ dx\]
\[=\ln(x^22x+10)+\dfrac 13(\tan^{1}(\frac{x1}{3}))+C\] – Rishabh Cool · 10 months ago
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PROBLEM 2:
\[\int_0^1 \dfrac{\ln \dfrac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]
This problem was solved by Samarth Agarwal. – Nihar Mahajan · 10 months ago
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\(\ln\dfrac{(x+a)^{x+a}}{(x+b)^{x+b}}=(x+a)\ln(x+a)  (x+b)\ln(x+b)\)
\(\dfrac{\ln\dfrac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln(x+a)\ln(x+b)}=\dfrac{1}{(x+b)\ln(x+b)}  \dfrac{1}{(x+a)\ln(x+a)}\)
now to evaluate \(\displaystyle \int \dfrac{dx}{(x+k)\ln(x+k)} \)
\(\ln(x+k)=t\)
\(\dfrac{1}{x+k} dx =dt\)
\(\therefore the\ integral\ is\ \dfrac{dt}{t} = \ln(t) = \ln(x+t)\)
Using this and putting limits we get the answer as :
\(\ln\dfrac{\ln(1+b)\ln(a)}{\ln(1+a)\ln(b)}\) – Samarth Agarwal · 10 months ago
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Problem 13
\[I_n=\int_{\pi}^{\pi}\left(\dfrac{\sin nx }{(1+\pi^x)\sin x}\right)\mathrm{d}x\] \(n=0,1,2,....\)
\[I_{n+2}I_n=k\times 100!, k\in \mathbf{Z}\]
Find \(k!\)
This problem was solved by Nihar Mahajan and later by Vighnesh Shenoy – Rishabh Cool · 10 months ago
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Note that \(I_{n+2}  I_n = \displaystyle\int_{\pi}^{\pi} \dfrac{\sin (nx+2x)  \sin nx}{(1+\pi^{x})\sin x} \ dx = \displaystyle\int_{\pi}^{\pi} \dfrac{2\sin x \cos ((n+1)x)}{(1+\pi^{x})\sin x} \ dx = \int_{\pi}^{\pi} \dfrac{2 \cos ((n+1)x)}{(1+\pi^{x})} \ dx\)
Now since the integrand is an even function we use the integration trick \(\displaystyle\int_{a}^{a} \dfrac{E(x)}{1+\pi^x} \ dx = \displaystyle\int_{0}^{a} E(x) \ dx \) :
\[\int_{0}^{\pi} 2 \cos ((n+1)x) = \dfrac{2\sin(n+1)}{n+1} \bigg_{0}^{\pi} = 0\]
Hence , \(k=0\Rightarrow k!=1\) . – Nihar Mahajan · 10 months ago
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\( I_{n+2}  I_{n} = \displaystyle \int_{0}^{\pi} \dfrac{\sin((n+2)x)  \sin(nx) dx }{\sin x } \)
\( I_{n+2}  I_{n}= \displaystyle \int_{0}^{\pi} \dfrac{2\sin(x) \cos((n+1)x)dx}{\sin x } = 2\int_{0}^{\pi} \cos((n+1)x) dx = 0 \)
\( k = 0, k! = 1 \) – Vighnesh Shenoy · 10 months ago
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PROBLEM 10:
If,
\(S=\cos^3(x) \frac{1}{2}\cos^5(x)\frac{1}{3}\cos^7(x)\frac{1}{4}\cos^9(x)...\infty\)
Then find \(\displaystyle \int_{\frac{\pi}{2}}^{\frac{5\pi}{2}} S \mathrm{d}x\).
This problem was solved by Samarth Agarwal but Vishnu Bhagyanath will post the next question. – Akshay Yadav · 10 months ago
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Akshay's Approach:
Let me familiarize you with the series I used here,
\[\ln(1x^2)=\sum_{n=1}^{\infty} \frac{x^{2n}}{n}\]
You must take \(\cos x\) common and then the integral would become,
\[\displaystyle \int \cos(x)\ln(\sin^2 (x))\mathrm{d}x\] – Vishnu Bhagyanath · 10 months ago
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@Akshay Yadav – Samarth Agarwal · 10 months ago
Is the answer 0Log in to reply
Problem 5:
if \(f(x)f(\dfrac{1}{x})=f(x)+f (\dfrac{1}{x})\) and\( f(10)=1001\) , find \(f(5)\).
P.S.: this is an easy problem to initiate calculus problems other than integration.
This problem was solved by Vighnesh Shenoy – Samarth Agarwal · 10 months ago
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We get,
\( c^{2} = 2c \)
\( c= 2 \) or \( c = 0 \)
Both of these do not satisfy,
\( f(10) = 1001 \)
Now,
\( f(x) = \dfrac{f\left(\dfrac{1}{x}\right)}{f\left(\dfrac{1}{x}\right)  1} \)
\( \therefore \left( f(x)1 \right) \cdot \left ( f\left(\dfrac{1}{x}\right)  1 \right) = 1 \)
Let,
\( f(x) = g(x) + 1 \rightarrow f\left(\dfrac{1}{x}\right) = g\left(\dfrac{1}{x}\right) + 1 \)
\( \therefore g(x) \cdot g\left(\dfrac{1}{x}\right) = 1 \)
\( g(x) \) is a polynomial of the type \( \pm x^{n} \)
\( \therefore f(x) = 1 \pm x^{n} \)
Substitute x = 10,
\( 1001 = 1 \pm 10^{n} \)
\( \pm 10^{n} = 1000 \)
\( 10^{n} \) can not be negative.
\( \therefore 10^{n} = 1000 \rightarrow n = 3 \)
\( f(x) = x^{3} + 1 \)
\( f(5) = 125 + 1 = 126 \)
I remember our sir discussing this specific type of function once in class. I also feel this is more of algebra rather than calculus. – Vighnesh Shenoy · 10 months ago
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PROBLEM 14:
Suppose \(a,b\) are real numbers such that \(a+b=1\). Then prove that the minimum value of the integral \( \displaystyle\int_{0}^{\pi}(a\sin x+b\sin 2x)^{2}\ dx \) is \(\dfrac{\pi}{4}\) and it occurs at \(a=b=\dfrac{1}{2}\). (Use Calculus only).
This problem was solved by Vighnesh Shenoy. – Nihar Mahajan · 10 months ago
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\( I = \displaystyle \int_{0}^{\dfrac{\pi}{2}} (a\sin x + b \sin 2x )^{2} + (a \sin x  b \sin 2x)^{2} dx \)
\( \therefore I = \displaystyle 2\int_{0}^{\dfrac{\pi}{2}} a^{2} \sin^{2}x + b^{2} \sin^{2}2x dx \)
On the first integral use, \( \displaystyle \int_{0}^{2a} f(x) dx = \int_{0}^{a} f(x) + f(2ax) dx \)
\( I =\displaystyle 2\int_{0}^{\dfrac{\pi}{4}}a^{2}dx + 4\int_{0}^{\dfrac{\pi}{4}}b^{2} \sin^{2}2x dx \)
Use that property again on the second integral,
\( I = 2\dfrac{a^{2}\pi}{4} + 4\dfrac{b^{2}\pi}{8} = \pi \cdot \dfrac{a^{2} + b^{2}}{2} = \pi \times \dfrac{(a^{2} + (1a)^{2})}{2} \)
Differentiate with respect to a,
\( \dfrac{dI}{da} = \pi \times \dfrac{2a 2(1a)}{2} = 0 \rightarrow a = \dfrac{1}{2} = b \)
Differentiate again with respect to a,
\( \dfrac{d^{2}I}{da^{2}} =\pi \times 2 > 0 \)
Thus the value is minimum, and occurs at \( a = b = \dfrac{1}{2} \)
\( I_{min} = \dfrac{\pi}{2} \times \left( \left(\dfrac{1}{2}\right)^{2} \right) \times 2 = \dfrac{\pi}{4} \) – Vighnesh Shenoy · 10 months ago
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1) How did you get second step from first?
2) How is \( \displaystyle \int_{0}^{2a} f(x) dx = \int_{0}^{a} f(x) + f(2ax) dx \) ?
BTW Nice solution , Post next problem vighu. – Nihar Mahajan · 10 months ago
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Proof :
\( I = \displaystyle \int_{0}^{2a}f(x)dx = \int_{0}^{a}f(x)dx + \int_{a}^{2a}f(x)dx \)
In the second integral,
\( 2a  x = t, \rightarrow dx =  dt \)
\( \displaystyle \int_{a}^{2a}f(x)dx = \int_{a}^{0}f(2at)dt = \int_{0}^{a}f(2at)dt = \int_{0}^{2a}f(2ax)dx \)
\( \therefore I = \displaystyle \int_{0}^{a}f(x) + f(2ax)dx \) – Vighnesh Shenoy · 10 months ago
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– Nihar Mahajan · 10 months ago
Thanks!Log in to reply
Find the general solution to the differential equation,
\( \dfrac{dy}{dx}  2y\tan(x )= \sin(2x) \)
This problem was solved by Rishabh Cool. – Vighnesh Shenoy · 10 months ago
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\[y\cos^2 x=\int 2\sin x \cos^3 xdx\] Make substitution \(\cos x=t\) to evaluate the integral such that \(\sin x dx=dt\) \[\boxed{y\cos ^2 x=\dfrac{\cos^4 x}{2}+C}\] – Rishabh Cool · 10 months ago
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PROBLEM 11:
\[ \displaystyle \int_{0}^{\infty} e^{\sqrt{x}}\mathrm{d}x\]
This problem was solved by Vighnesh Shenoy. – Vishnu Bhagyanath · 10 months ago
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\( I = \displaystyle 2 \int_{0}^{\infty}te^{t}dt \)
\( \displaystyle \int te^{t}dt = e^{t}(t+1) \)
\( I = 2\left[ e^{t}(t+1) \right]_{0}^{\infty} = 2\left( \displaystyle \lim_{x \rightarrow \infty} e^{t}(t+1)  1 \right) \) = 2
I did not use the gamma function on purpose as it is prohibited. – Vighnesh Shenoy · 10 months ago
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PROBLEM 7:
Find all real numbers \(x\) such that \(\displaystyle\int_0^x t^2\sin (xt)\ dt=x^2\)
This problem has been solved by Akshay Yadav. – Nihar Mahajan · 10 months ago
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\(\int_0^{x} t^2 \sin(xt) \mathrm{d}t=x^2\)
\(\int_0^{x} t^2[ \sin(x)\cos(t)\sin(t)\cos(x)] \mathrm{d}t=x^2\)
\(\int_0^{x} t^2 \sin(x)\cos(t)\mathrm{d}t \int_0^{x} t^2 \sin(t)\cos(x) \mathrm{d}t=x^2\)
Applying linearity and using Integration by parts,
\(\sin(x)\int_0^{x} t^2 \cos(t)\mathrm{d}t\cos(x) \int_0^{x} t^2 \sin(t)\mathrm{d}t=x^2\)
\(\sin(x)[x^2 \sin (x) 2\sin(x)+2x\cos (x)]\cos(x)[x^2\cos(x)+2x\sin(x)+2\cos(x) 2]=x^2\)
\(x^2+2\cos(x)2=x^2\)
\(\cos(x)=1\)
\(x=2n\pi \text{ } \forall n \in \mathrm{I}\) – Akshay Yadav · 10 months ago
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Problem 22
\[\displaystyle \dfrac{\int_{0}^{\pi} x\ln \sin x \ dx}{\int_{0}^{\pi} \ln\sin x \ dx} =\pi/2\]
This problem was first solved by Rishabh Cool. – Harsh Shrivastava · 10 months ago
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\[I_1=\int_{0}^{\pi} x\ln \sin x \ dx\]
\[=\int_0^{\pi} (\pix)\ln \sin (\pix) \ dx\]
\[=\int_{0}^{\pi} \pi\ln \sin x \ dxI_1\]
\[\implies I_1=\dfrac{\pi}{2}\int_{0}^{\pi} \ln \sin x \ dx=\dfrac{\pi}{2}I_2\]
And thus :
\[\dfrac{I_1}{I_2}=\dfrac{\pi}{2}=I\]
Hence proved... – Rishabh Cool · 10 months ago
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– Adarsh Kumar · 10 months ago
Post the next question please.Log in to reply
Problem 21 by Adarsh Kumar:\[\int[(x+a)^{3}(x+b)^{5}]^{\frac{1}{4}}dx=?\]
This problem was first solved by Harsh Shrivastva. – Akshay Yadav · 10 months ago
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PROBLEM 20
prove that
\(\displaystyle\int _{ a }^{ b }{ \frac { x\quad dx }{ \sqrt { (xa)(bx) } } } =\frac { \pi (a+b) }{ 2 } \)
This problem was solved first by Adarsh Kumar and then by others. – Hummus A · 10 months ago
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\[I=\int_a^b \frac {(a+b x)dx }{ \sqrt { (xa)(bx) } }\]
\[=\int_a^b\dfrac{(a+b)dx}{ \sqrt { (xa)(bx) } }I\]
\[\implies I=\int_a^b \dfrac{(a+b)}{2}(\dfrac{dx}{ \sqrt { (xa)(bx) } })\]
\[=\int_a^b\dfrac{a+b}{2}(\dfrac{dx}{ \sqrt {\dfrac{(ba)^2}{2} (x(\dfrac{(a+b)}{2}))^2} })\]
Now using \(\int \dfrac{dx}{\sqrt{a^2x^2}}=\sin^{1} \dfrac xa\).
\[I=\dfrac{(a+b)}{2}(\sin^{1}(\dfrac{2x(a+b)}{ba}))_a^b\]
\[\large I=\boxed{\dfrac{\pi(a+b)}{2}}\] – Rishabh Cool · 10 months ago
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Very beautiful problem! Firstly, using \(\int_{a}^{b} f(x) \ dx = \int_{a}^{b} f(a+bx) \ dx\) , we have:
\[I=\int_{a}^{b} \dfrac{x}{\sqrt{(xa)(bx)}} \ dx = \int_{a}^{b} \dfrac{a+bx}{\sqrt{(xa)(bx)}} \ dx\]
Adding both we get \(2I= \displaystyle\int_{a}^{b}\dfrac{a+b}{\sqrt{(xa)(bx)}} \ dx\) . Now we need to get rid of mutiple variable of denominator by introducing single new variable. So substitute \(y=\dfrac{xa}{ba}\) and note that \(1y = \dfrac{bx}{ba}\) and \((ba)dy=dx\) and the integral changes to: \(I=\dfrac{a+b}{2} \displaystyle\int_{0}^{1} \dfrac{1}{\sqrt{y(1y)}} \ dy \) which is simply the antiderivative for \(\arcsin\). So we have the integral evaluated as \(\dfrac{\pi(a+b)}{2}\) . – Nihar Mahajan · 10 months ago
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\(\displaystyle \int_a^b \frac{1}{\sqrt{xbx^2ab+ax}} \mathrm{d}x\)
\(\displaystyle \int_a^b \frac{1}{\sqrt{\left(x\frac{a+b}{2}\right)^2+\left(\frac{ab}{2}\right)^2}} \mathrm{d}x\)
\(\arcsin \left(\frac{2x}{ab}\frac{a+b}{ab}\right)_a^b\)
\(\arcsin (1)\arcsin (1) \rightarrow \pi\) – Akshay Yadav · 10 months ago
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– Adarsh Kumar · 10 months ago
Solution to problem 20:\[\int_a^b\dfrac{x dx}{\sqrt{(xa)(bx)}}\]\[=\int_a^b\dfrac{(a+bx)dx}{\sqrt{(xa)(bx)}}\]\[\Longrightarrow 2I=(a+b)\int_a^b(a+b)\dfrac{dx}{\sqrt{(xa)(bx)}}\].Now substituting \[z=\dfrac{a+b}{2}x\],and changing the limits accordingly,\[2I=(a+b)\int_{\dfrac{ba}{2}}^{\dfrac{ab}{2}}\dfrac{dz}{\sqrt{(\dfrac{ba}{2}z)(\dfrac{ba}{2}+z)}}\]\[\int \dfrac{dx}{\sqrt{a^2x^2}}=\sin^{1}\dfrac{x}{a}\],doing the calculations we get,\[2I=(a+b)(\pi)\\ \Longrightarrow I=\dfrac{(a+b)\pi}{2}\].It is done!Log in to reply
– Nihar Mahajan · 10 months ago
You post the next question ;)Log in to reply
– Hummus A · 10 months ago
WOW!, 4 people at the same timeLog in to reply
– Samarth Agarwal · 10 months ago
Another way is to take sqrt(xa)=t and solve the trivial integralLog in to reply
PROBLEM 19:
Find the quadratic mean and the arithmetical mean of the function \[y=A_1\sin (x)+A_3\sin (3x)\]
Subsequently find the minimum value of quadratic mean as \(A_1,A_3 \in \mathbb{R}\).
This question was solved by Hummus A. – Akshay Yadav · 10 months ago
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quadratic mean: first we square the function,we get
\({ A }_{ 1 }^{ 2 }\sin ^{ 2 }{ x } +2{ A }_{ 1 }{ A }_{ 3 }\sin { (x) } \sin { (3x) } +{ A }_{ 3 }^{ 2 }\sin ^{ 2 }{ (3x) } \)
since all the functions here are periodic,the RMS over all time is the RMS over one period of the function
so computing the RMS we get
\(\sqrt { \frac { 1 }{ \pi } \displaystyle\int _{ 0 }^{ \pi }{ { A }_{ 1 }^{ 2 }\sin ^{ 2 }{ x } +2{ A }_{ 1 }{ A }_{ 3 }\sin { (x) } \sin { (3x) } +{ A }_{ 3 }^{ 2 }\sin ^{ 2 }{ (3x) } dx } } =\\ \\ \sqrt { \frac { 1 }{ \pi } ({ A }_{ 1 }^{ 2 }\displaystyle\int _{ 0 }^{ \pi }{ \sin ^{ 2 }{ x } dx } +2{ A }_{ 1 }{ A }_{ 3 }\displaystyle\int _{ 0 }^{ \pi }{ \sin { (x)\sin { (3x)dx } +{ A }_{ 3 }^{ 2 } } \displaystyle\int _{ 0 }^{ \pi }{ \sin ^{ 2 }{ (3x) } dx } }) } =\sqrt { \frac { { A }_{ 1 }^{ 2 } }{ 2 } +\frac { { A }_{ 3 }^{ 2 } }{ 2 } } \)
the minimum value of the quadratic mean here is 0,when \({ A }_{ 1 }={ A }_{ 3 }=0\),otherwise it has no minimum
mean:
this is the integral of the function over the period divided by the the difference of the bounds,which is
\(\Large\frac { \displaystyle\int _{ \pi /2 }^{ 3\pi /2 }{ \sin { x } +\sin { 3x } dx } }{ 2\pi } =\frac { 0 }{ 2\pi } =0\)
i'm not sure if this solution is error free since i was too lazy to get pen and paper to work on it and did it mostly mentally,so feel free to notify me about any errors :) – Hummus A · 10 months ago
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Problem 17:
\(f(x) = \sqrt{x  \sqrt{12ax36a^{2}}}\)
Find the value of \(\displaystyle \int f(x)dx\)
This problem was solved by Nihar Mahajan. – Harsh Shrivastava · 10 months ago
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Note that \(f(x) = \sqrt{\left(\sqrt{x3a}\right)^2 + \left(\sqrt{3a}\right)^2  2\sqrt{3ax9a^2}} = \sqrt{(\sqrt{x3a}  \sqrt{3a})^2} = \sqrt{x3a}\sqrt{3a}\).
Hence, \(\displaystyle\int \sqrt{x3a}\sqrt{3a} = \dfrac{2(x3a)^{3/2}}{3}  \sqrt{3a}x+C\) . – Nihar Mahajan · 10 months ago
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PROBLEM 15:
Evaluate the following integral,
\( \displaystyle \int_{0}^{\frac{\pi}{2}} \dfrac{4a\sin^{2}x + 6b^{2}\cos^{2}x}{(a^{2}\sin^{2}x + b^{3}\cos^{2}x)^{2}}dx \)
This problem has been solved by Adarsh Kumar. – Vighnesh Shenoy · 10 months ago
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– Adarsh Kumar · 10 months ago
\[\dfrac{I}{2}=\int_{0}^{\frac{\pi}{2}}\dfrac{2a\sin^2x}{(a^2\sin^2x+b^3\cos^2x)^2}dx+\int_{0}^{\frac{\pi}{2}}\dfrac{3b^2\cos^2x}{(a^2\sin^2x+b^3\cos^2x)^2}dx\]\[I_1(a)=\int_{0}^{\frac{\pi}{2}}\dfrac{dx}{a^2\sin^2x+b^3\cos^2x}\]\[\Longrightarrow I_1'(a)=\int_{0}^{\frac{\pi}{2}}\dfrac{dx}{(a^2\sin^2x+b^3\cos^2x)^2}\times 2a\sin^2x\],the same definition goes for \(I_2(b)\),and hence \[I_2'(b)=\int_0^{\frac{\pi}{2}}\dfrac{dx}{(a^2\sin^2x+b^3\cos^2x)^2}\times 3b^2\cos^2x\],hence,\[I_1'(a)+I_2'(b)=\dfrac{I}{2}(*)\],evaluating \(I_1(a)\) and \(I_2(b)\) using \(\tan\) and \(\sec\) technique and using the fact that \(\int\dfrac{dx}{x^2+a^2}=\dfrac{1}{a}\tan^{1}{x}\),we finally get that,\[I_1'(a)=\dfrac{1}{a^2b^{\frac{3}{2}}}\dfrac{\pi}{2}\\ \text{and}\ I_{2}'(b)=\dfrac{3}{2}\dfrac{1}{ab^{\frac{5}{2}}}\dfrac{\pi}{2}\].Substituting these values in \(*\) we get that \[I=\pi\dfrac{3a+2b}{2a^2b^{\frac{5}{2}}}\],it is done!Log in to reply
PROBLEM 9:
Find the minimum area of the region bounded by the curve \(y=a^3x^2a^4x\) and the line \(y=x\) where \(a>0\) .
This problem has been solved by Akshay Yadav. – Nihar Mahajan · 10 months ago
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I won't be providing a rigorous solution to problem as it is very long however here is what I did,
I figured the point of intersection of the two curves in terms of \(a\).
I transformed the two curves that the area we need to calculate remains positive, (I am unable to provide an image of graph because of LaTeX, perhaps some one can help me).
Then integration to find area and subtraction, you will get area as a function of \(a\).
Differentiate it and find the global minima. – Akshay Yadav · 10 months ago
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@Akshay Yadav post next question, I'm not taking part, just thought I'd pitch in
Vighnesh's Observation:
Instead of integrating and differentiating you could have used this,
\( \displaystyle \dfrac{d}{da} \left( \int_{g(a)}^{h(a)} f(x)dx \right) = f(h(a))h'(a)  f(g(a))g'(a) \) to find the value of a first. Finding the value of a first makes substitution of limits easier later. – Vishnu Bhagyanath · 10 months ago
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PROBLEM 8:
Find the length of an arc of curve
\[y=\frac{a}{2}\left(e^{\frac{x}{a}}+e^{\frac{x}{a}}\right)\]
between \(x=0\) and \(x=a\).
Answer in terms of \(a\) and \(e\) in simplest form.
This problem has been solved by Nihar Mahajan. – Akshay Yadav · 10 months ago
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Note that \(y=a\cosh\left(\dfrac{x}{a}\right) \Rightarrow \dfrac{dy}{dx} = \sinh\left(\dfrac{x}{a}\right)\) (Using chain rule) . Now,
\[\sqrt{1+\left(\dfrac{dy}{dx}\right)^2} = \sqrt{1+\sinh^2\left(\dfrac{x}{a}\right)} = \left \cosh\left(\dfrac{x}{a}\right) \right \]
Now length of the arc from \(x=0\) to \(x=a\) is given by the integral: \(\displaystyle\int_{0}^{a} \cosh\left(\dfrac{x}{a}\right) \ dx\)
Substitute \(y=\dfrac{x}{a} \Rightarrow dx=a \ dy\) and changing limits , the integral becomes \(a\left[\sinh(y)\right]\bigg_{0}^{1} = a\sinh(1) = \dfrac{a(e^21)}{2e}\) – Nihar Mahajan · 10 months ago
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PROBLEM 6:
Find the closed form of the indefinite integral,
\( \displaystyle \int \dfrac{dx}{1\cot x} \)
This problem has been solved by Nihar Mahajan – Vighnesh Shenoy · 10 months ago
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\[ \displaystyle \int \dfrac{dx}{1\cot x} = \dfrac{1}{2} \int \dfrac{2\sin x}{\sin x  \cos x} \ dx=\dfrac{1}{2} \int \dfrac{\sin x + \cos x + \sin x  \cos x}{\sin x  \cos x} \ dx \\ = \dfrac{1}{2} \left[\int 1 \ dx + \int \dfrac{\sin x + \cos x}{\sin x  \cos x} \ dx \right] \]
Now substituting \(u=\sin x  \cos x \Rightarrow du = \cos x+\sin x \ dx\) , the above integral changes to:
\[\dfrac{1}{2}\left[x\int \dfrac{1}{u} \ du\right] = \dfrac{x\ln(\cos x  \sin x)}{2} + C\] – Nihar Mahajan · 10 months ago
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New note has been published. Click here to access it. – Akshay Yadav · 10 months ago
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@Harsh Shrivastava @Adarsh Kumar @Vighnesh Shenoy @Samarth Agarwal @Vishnu Bhagyanath @Hummus a and others shift to second note. – Nihar Mahajan · 10 months ago
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BONUS PROBLEM: Evaluate \(\int y^{3/4}(y1)^{5/4} \ dy \) . (Actually its my doubt :P)
This problem is not a part of the contest, however it was first solved by Rishabh Cool. – Nihar Mahajan · 10 months ago
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Thus integral transforms to: \[\int t^{\frac{5}{4}}dt\] And you are done.. I guess :) – Rishabh Cool · 10 months ago
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– Nihar Mahajan · 10 months ago
Thanks! I have solved Problem 21 of this contest with your help now ;)Log in to reply
– Nihar Mahajan · 10 months ago
Guys, please help me with this problem...Log in to reply
PROBLEM 18:
Let \(\alpha \ , \ \beta\) be the distinct positive roots of the equation \(2x=\tan x\) . Then prove that:
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx = 0\]
This question was solved by Akshay Yadav. – Nihar Mahajan · 10 months ago
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\[\displaystyle \int_0^{1} \frac{\cos((\beta+\alpha)x)\cos((\beta\alpha)x)}{2}\mathrm{d}x\]
\[\displaystyle \int_0^{1} \frac{\cos((\beta+\alpha)x)}{2}\mathrm{d}x \displaystyle \int_0^{1} \frac{\cos((\beta\alpha)x)}{2}\mathrm{d}x\]
Applying linearity,
\[\frac{1}{2} \displaystyle \int_0^{1} \cos((\beta+\alpha)x)\mathrm{d}x+\frac{1}{2} \displaystyle \int_0^{1} \cos((\beta\alpha)x)\mathrm{d}x\]
Solving each integral we get,
\[\dfrac{\alpha\cos(\alpha)\sin(\beta)\sin(\alpha)\beta\cos(\beta)}{\beta^2\alpha^2}\]
\[\dfrac{\cos(\alpha)\cos(\beta)(\alpha\tan(\beta)\tan(\alpha)\beta)}{\beta^2\alpha^2}\]
Now,
\(\tan(\alpha)=2\alpha\) and \(\tan(\beta)=2\beta\)
So,
\[\dfrac{\cos(\alpha)\cos(\beta)(\alpha(2\beta)(2\alpha)\beta)}{\beta^2\alpha^2}\]
Hence its equal to \(\boxed{0}\). – Akshay Yadav · 10 months ago
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– Adarsh Kumar · 10 months ago
Great!Post the next question!Log in to reply
PROBLEM 16:
\[\int\dfrac{dx}{(5+4\cos x)^2}=?\].
This problem was solved by Harsh Shrivastava. – Adarsh Kumar · 10 months ago
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\(I = \displaystyle 2\int \dfrac{1+t^{2}}{(9+t^{2})^2}dt\)
\(I= \displaystyle 2\int\dfrac{1}{(9+t^{2})^2} dt2\int \dfrac{t^2}{(9+t^{2})^2}dt\)
In both the integrals,substitute \(t = 3\tan p\),
Both the integrals will now be trivial to evaluate.
Sorry for this concise solution without any elaboration 'coz I am not in my home and using latex on mobile is very(!) cumbersome+tedious.
I will improve this solution when I'll reach home. – Harsh Shrivastava · 10 months ago
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