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:
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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..)
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Solution to problem 4:
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 \(4\sqrt34\dfrac{\pi}{3}\)
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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.
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SOLUTION TO PROBLEM 1:
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.
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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.
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SOLUTION TO PROBLEM 2:
\(\ln\dfrac{(x+a)^{x+a}}{(x+b)^{x+b}}=(x+a)\ln(x+a)  (x+b)\ln(x+b)\)
\(\dfrac{\ln\dfrac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln(x+a)\ln(x+b)}=\dfrac{1}{(x+b)\ln(x+b)}  \dfrac{1}{(x+a)\ln(x+a)}\)
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 :
\(\ln\dfrac{\ln(1+b)\ln(a)}{\ln(1+a)\ln(b)}\)
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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.
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SOLUTION TO PROBLEM 3:
\[\displaystyle \int \dfrac{2(x1)+1}{x^22x+10} \ dx\]
\[=\displaystyle \int \dfrac{2(x1)}{x^22x+10}+\dfrac{1}{(x1)^2+9} \ dx\]
\[=\ln(x^22x+10)+\dfrac 13(\tan^{1}(\frac{x1}{3}))+C\]
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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
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Let us consider f(x) is a constant c.
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.
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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.
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Area under the graph of \(\cos^{2n1} (x) \) from \(k \text{ to } 2 \pi + k \) is zero as the negative half of the graph cancels out with the positive half.
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,
\[\displaystyle \int \cos(x)\ln(\sin^2 (x))\mathrm{d}x\]
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Is the answer 0 @Akshay Yadav
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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
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SOLUTION TO PROBLEM 13:
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 , \(k=0\Rightarrow k!=1\) .
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\( I_{n} = \displaystyle \int_{\pi}^{\pi} \dfrac{\sin(nx)dx}{(1+\pi^{x})\sin(x)} = \int_{0}^{\pi} \dfrac{\sin(nx)dx}{\sin(x)} \)
\( I_{n+2}  I_{n} = \displaystyle \int_{0}^{\pi} \dfrac{\sin((n+2)x)  \sin(nx) dx }{\sin x } \)
\( I_{n+2}  I_{n}= \displaystyle \int_{0}^{\pi} \dfrac{2\sin(x) \cos((n+1)x)dx}{\sin x } = 2\int_{0}^{\pi} \cos((n+1)x) dx = 0 \)
\( k = 0, k! = 1 \)
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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.
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SOLUTION TO PROBLEM 7:
\(\int_0^{x} t^2 \sin(xt) \mathrm{d}t=x^2\)
\(\int_0^{x} t^2[ \sin(x)\cos(t)\sin(t)\cos(x)] \mathrm{d}t=x^2\)
\(\int_0^{x} t^2 \sin(x)\cos(t)\mathrm{d}t \int_0^{x} t^2 \sin(t)\cos(x) \mathrm{d}t=x^2\)
Applying linearity and using Integration by parts,
\(\sin(x)\int_0^{x} t^2 \cos(t)\mathrm{d}t\cos(x) \int_0^{x} t^2 \sin(t)\mathrm{d}t=x^2\)
\(\sin(x)[x^2 \sin (x) 2\sin(x)+2x\cos (x)]\cos(x)[x^2\cos(x)+2x\sin(x)+2\cos(x) 2]=x^2\)
\(x^2+2\cos(x)2=x^2\)
\(\cos(x)=1\)
\(x=2n\pi \text{ } \forall n \in \mathrm{I}\)
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PROBLEM 11:
\[ \displaystyle \int_{0}^{\infty} e^{\sqrt{x}}\mathrm{d}x\]
This problem was solved by Vighnesh Shenoy.
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\( x = t^{2} \rightarrow dx = 2tdt \)
\( I = \displaystyle 2 \int_{0}^{\infty}te^{t}dt \)
\( \displaystyle \int te^{t}dt = e^{t}(t+1) \)
\( I = 2\left[ e^{t}(t+1) \right]_{0}^{\infty} = 2\left( \displaystyle \lim_{x \rightarrow \infty} e^{t}(t+1)  1 \right) \) = 2
I did not use the gamma function on purpose as it is prohibited.
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Find the general solution to the differential equation,
\( \dfrac{dy}{dx}  2y\tan(x )= \sin(2x) \)
This problem was solved by Rishabh Cool.
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Integration factor =\(e^{\int 2\tan x dx}=\cos^2 x\).
\[y\cos^2 x=\int 2\sin x \cos^3 xdx\] Make substitution \(\cos x=t\) to evaluate the integral such that \(\sin x dx=dt\) \[\boxed{y\cos ^2 x=\dfrac{\cos^4 x}{2}+C}\]
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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.
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\( I = \displaystyle \int_{0}^{\pi} (a\sin x + b\sin 2x)^{2} dx \)
\( I = \displaystyle \int_{0}^{\dfrac{\pi}{2}} (a\sin x + b \sin 2x )^{2} + (a \sin x  b \sin 2x)^{2} dx \)
\( \therefore I = \displaystyle 2\int_{0}^{\dfrac{\pi}{2}} a^{2} \sin^{2}x + b^{2} \sin^{2}2x dx \)
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 \( a = b = \dfrac{1}{2} \)
\( I_{min} = \dfrac{\pi}{2} \times \left( \left(\dfrac{1}{2}\right)^{2} \right) \times 2 = \dfrac{\pi}{4} \)
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I have few doubts, doubt killer:
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.
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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,
\( 2a  x = t, \rightarrow dx =  dt \)
\( \displaystyle \int_{a}^{2a}f(x)dx = \int_{a}^{0}f(2at)dt = \int_{0}^{a}f(2at)dt = \int_{0}^{2a}f(2ax)dx \)
\( \therefore I = \displaystyle \int_{0}^{a}f(x) + f(2ax)dx \)
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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
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SOLUTION TO PROBLEM 6: When we convert \(\cot x\) in terms of \(\sin x\) and \(\cos x\) , we get:
\[ \displaystyle \int \dfrac{dx}{1\cot x} = \dfrac{1}{2} \int \dfrac{2\sin x}{\sin x  \cos x} \ dx=\dfrac{1}{2} \int \dfrac{\sin x + \cos x + \sin x  \cos x}{\sin x  \cos x} \ dx \\ = \dfrac{1}{2} \left[\int 1 \ dx + \int \dfrac{\sin x + \cos x}{\sin x  \cos x} \ dx \right] \]
Now substituting \(u=\sin x  \cos x \Rightarrow du = \cos x+\sin x \ dx\) , the above integral changes to:
\[\dfrac{1}{2}\left[x\int \dfrac{1}{u} \ du\right] = \dfrac{x\ln(\cos x  \sin x)}{2} + C\]
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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.
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SOLUTION TO PROBLEM 8:
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 \(a\left[\sinh(y)\right]\bigg_{0}^{1} = a\sinh(1) = \dfrac{a(e^21)}{2e}\)
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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.
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SOLUTION TO PROBLEM 9:
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.
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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.
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\[\dfrac{I}{2}=\int_{0}^{\frac{\pi}{2}}\dfrac{2a\sin^2x}{(a^2\sin^2x+b^3\cos^2x)^2}dx+\int_{0}^{\frac{\pi}{2}}\dfrac{3b^2\cos^2x}{(a^2\sin^2x+b^3\cos^2x)^2}dx\]\[I_1(a)=\int_{0}^{\frac{\pi}{2}}\dfrac{dx}{a^2\sin^2x+b^3\cos^2x}\]\[\Longrightarrow I_1'(a)=\int_{0}^{\frac{\pi}{2}}\dfrac{dx}{(a^2\sin^2x+b^3\cos^2x)^2}\times 2a\sin^2x\],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!
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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.
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SOLUTION TO PROBLEM 17:
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, \(\displaystyle\int \sqrt{x3a}\sqrt{3a} = \dfrac{2(x3a)^{3/2}}{3}  \sqrt{3a}x+C\) .
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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.
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here's my solution
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 :)
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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.
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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!
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WOW!, 4 people at the same time
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You post the next question ;)
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First simplifying \(\displaystyle \int_a^b \frac{1}{\sqrt{(xa)(bx)}} \mathrm{d}x\), I would be providing beginning steps only as its very long generalization.
\(\displaystyle \int_a^b \frac{1}{\sqrt{xbx^2ab+ax}} \mathrm{d}x\)
\(\displaystyle \int_a^b \frac{1}{\sqrt{\left(x\frac{a+b}{2}\right)^2+\left(\frac{ab}{2}\right)^2}} \mathrm{d}x\)
\(\arcsin \left(\frac{2x}{ab}\frac{a+b}{ab}\right)_a^b\)
\(\arcsin (1)\arcsin (1) \rightarrow \pi\)
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SOLUTION TO PROBLEM 20:
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 \(\dfrac{\pi(a+b)}{2}\) .
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First using property \(\int_a^b f(x)dx=\int f(a+bx)dx\) and then adding to simplify the integral as:
\[I=\int_a^b \frac {(a+b x)dx }{ \sqrt { (xa)(bx) } }\]
\[=\int_a^b\dfrac{(a+b)dx}{ \sqrt { (xa)(bx) } }I\]
\[\implies I=\int_a^b \dfrac{(a+b)}{2}(\dfrac{dx}{ \sqrt { (xa)(bx) } })\]
\[=\int_a^b\dfrac{a+b}{2}(\dfrac{dx}{ \sqrt {\dfrac{(ba)^2}{2} (x(\dfrac{(a+b)}{2}))^2} })\]
Now using \(\int \dfrac{dx}{\sqrt{a^2x^2}}=\sin^{1} \dfrac xa\).
\[I=\dfrac{(a+b)}{2}(\sin^{1}(\dfrac{2x(a+b)}{ba}))_a^b\]
\[\large I=\boxed{\dfrac{\pi(a+b)}{2}}\]
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Problem 21 by Adarsh Kumar:\[\int[(x+a)^{3}(x+b)^{5}]^{\frac{1}{4}}dx=?\]
This problem was first solved by Harsh Shrivastva.
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Solution 21 by Harsh Shrivastava:
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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.
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\[I=\dfrac{\overbrace{\int_{0}^{\pi} x\ln \sin x \ dx}^{\color{red}{I_1}}}{\underbrace{\int_{0}^{\pi} \ln\sin x \ dx}_{\color{red}{I_2}}}\]
\[I_1=\int_{0}^{\pi} x\ln \sin x \ dx\]
\[=\int_0^{\pi} (\pix)\ln \sin (\pix) \ dx\]
\[=\int_{0}^{\pi} \pi\ln \sin x \ dxI_1\]
\[\implies I_1=\dfrac{\pi}{2}\int_{0}^{\pi} \ln \sin x \ dx=\dfrac{\pi}{2}I_2\]
And thus :
\[\dfrac{I_1}{I_2}=\dfrac{\pi}{2}=I\]
Hence proved...
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Post the next question please.
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PROBLEM 16:
\[\int\dfrac{dx}{(5+4\cos x)^2}=?\].
This problem was solved by Harsh Shrivastava.
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Substituting \(\cos x = \frac{1t^{2}}{1+t^{2}} \), where \(t = \tan(x/2)\),the integral transforms into
\(I = \displaystyle 2\int \dfrac{1+t^{2}}{(9+t^{2})^2}dt\)
\(I= \displaystyle 2\int\dfrac{1}{(9+t^{2})^2} dt2\int \dfrac{t^2}{(9+t^{2})^2}dt\)
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.
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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.
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The integral given can be written as,
\[\displaystyle \int_0^{1} \frac{\cos((\beta+\alpha)x)\cos((\beta\alpha)x)}{2}\mathrm{d}x\]
\[\displaystyle \int_0^{1} \frac{\cos((\beta+\alpha)x)}{2}\mathrm{d}x \displaystyle \int_0^{1} \frac{\cos((\beta\alpha)x)}{2}\mathrm{d}x\]
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 \(\boxed{0}\).
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Great!Post the next question!
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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.
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Write it as : \[\int (\dfrac{y1}{y})^{\frac{5}{4}}\dfrac{dy}{y^2}\] Substitute \(\dfrac{y1}{y}=t\) such that \(\dfrac{dy}{y^2}=dt\).
Thus integral transforms to: \[\int t^{\frac{5}{4}}dt\] And you are done.. I guess :)
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Thanks! I have solved Problem 21 of this contest with your help now ;)
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Guys, please help me with this problem...
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New note has been published. Click here to access it.
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@Harsh Shrivastava @Adarsh Kumar @Vighnesh Shenoy @Samarth Agarwal @Vishnu Bhagyanath @Hummus a and others shift to second note.
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