Brilliant Integration Contest - Season 2 (Part 2)

Hi Brilliant! Due to many problems in part 1 it had become slow to load. So this is a sequel of Brilliant Integration - Season 2(Part 1)

The aims of the Integration contest are to improve skills in the computation of integrals, to learn from each other as much as possible, and of course to have fun. Anyone 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 integrals either definite or indefinite integrals.

• You are NOT allowed to post a multiple integrals problem.

• It is NOT compulsory to post original problems. But make sure it has not been posted on brilliant.

• Do not copy questions from last year's contest. If anyone found to do so he/she will be banned from taking further part in this contest

• You are also NOT allowed to post a solution using a contour integration or residue method.

The final answer can ONLY contain the following special functions: gamma function, beta function, Riemann zeta function, Dirichlet eta function, dilogarithm, digamma function, trigonometric integral, Wallis' integral, Bessel function, contour integration and Ramanujan's Master Theorem (including Mellin Transform).

Format your post is as follows:

 1 2 3 4 5 6 7 **SOLUTION OF PROBLEM xxx (number of problem) :** **[Post your solution here]** **PROBLEM xxx (number of problem) :** **[Post your problem here]** 

UPDATE:

• Tanishq Varshney has been banned from this contest indefinitely.

• Contour integration is allowed in the contest.

4 years, 2 months ago

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Problem 50:

Evaluate $\large \int_{0}^{\infty} \dfrac{\ln(1+x)}{\sqrt[4]{x^3}(1+x)}\, dx$

- 4 years, 2 months ago

Alternative method:

$\dfrac{\ln(1+x)}{1+x} = - \sum_{n=1}^{\infty} H_{n}(-x)^{n}$

Now use RMT. That's it. $\ddot \smile$

- 4 years, 2 months ago

Put $\dfrac{1}{1+x}=y$ to get :

$\displaystyle I = -\int _{ 0 }^{ 1 }{ { y }^{ (-1/4) }{ (1-y) }^{ -3/4 }\ln { (y) } dy }$

Use the definition of beta function we have :

$\displaystyle \int _{ 0 }^{ 1 }{ { y }^{ m-1 }{ (1-y) }^{ n-1 }dy } = \frac { \Gamma (m)\Gamma (n) }{ \Gamma (m+n) }$

Differentiating it with respect to $m$ we have :

$\displaystyle \frac { \Gamma (m)\Gamma (n) }{ \Gamma (m+n) } (\psi (m)-\psi (m+n)) = \int _{ 0 }^{ 1 }{ { y }^{ m-1 }{ (1-y) }^{ n-1 }\ln { (y) } dy }$

Put $\displaystyle m=\dfrac{3}{4} , n=\dfrac{1}{4}$ to get :

$\displaystyle I = \frac { \Gamma (3/4)\Gamma (1/4) }{ \Gamma (1) } (\psi (1)-\psi (3/4))$

Using reflection formula for the gamma function we have :

$\displaystyle \Gamma(1/4)\Gamma(1-1/4) = \dfrac{\pi}{\sin(\dfrac{\pi}{4})}=\sqrt{2}\pi$

We will use that $\displaystyle \psi(x+1) + \gamma = \int _{ 0 }^{ 1 }{ \frac { 1-{ t }^{ x } }{ 1-t } dt }$

$\displaystyle \Rightarrow \psi(x+1)-\psi(y+1) = \int _{ 0 }^{ 1 }{ \frac { { t }^{ y }-{ t }^{ x } }{ 1-t } dt }$

$\displaystyle \Rightarrow \psi(1)-\psi(3/4) = \int _{ 0 }^{ 1 }{ \frac { { t }^{ -1/4 }-1 }{ 1-t } dt }=J$

For evaluting $J$ we will put $t={x}^{4}$, to get :

$\displaystyle J = 4\int _{ 0 }^{ 1 }{ \frac { 1-x }{ 1-{ x }^{ 4 } } { x }^{ 2 }dx }$

$\displaystyle J = 4\int _{ 0 }^{ 1 }{ \frac { { x }^{ 2 } }{ (1+{ x }^{ 2 })(1+x) } dx }$

$\displaystyle J =2\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+x } } +\int _{ 0 }^{ 1 }{ \frac { 2x }{ 1+{ x }^{ 2 } } dx } -2\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 2 } } }$

$\displaystyle J = 3\ln { (2) }-\dfrac{\pi}{2}$

$\displaystyle \Rightarrow I = 3\sqrt { 2 } \pi \ln { (2) } -\frac { { \pi }^{ 2 } }{ \sqrt { 2 } }$

- 4 years, 2 months ago

Problem 49:

Evaluate $\large \displaystyle \int _{ 0 }^{ 1 }{ \ln { (\Gamma (x)) } dx. }$

This problem has been solved by Surya Prakesh.

- 4 years, 2 months ago

$I = \int_{0}^{1} \ln(\Gamma(x)) dx = \int_{0}^{1} \ln(\Gamma(1-t)) dt$

I used the substitution $x+t=1$. But from Euler's reflection formula for Gamma function i.e.

$\Gamma(x) \Gamma(1-x)= \pi \csc(\pi x) \implies \ln(\Gamma(1-x)) + \ln(\Gamma(x)) = \ln(\pi) + \ln(\csc(\pi x))$

\begin{aligned} I &= \int_{0}^{1} \ln(\Gamma(1-x)) dx\\ I &= -\int_{0}^{1} \ln(\Gamma(x)) dx + \int_{0}^{1} \ln(\pi) dx+ \int_{0}^{1} \ln(\csc(\pi x))dx \\ I &= -I + \ln(\pi) - \int_{0}^{1} \ln(\sin(\pi x))dx \\ 2I &= \ln(\pi) - \dfrac{1}{\pi} \int_{0}^{\pi} \ln(\sin(x))dx \\ 2I &= \ln(\pi) - \dfrac{2}{\pi} \int_{0}^{\pi / 2} \ln(\sin(x))dx \\ 2I &= \ln(\pi) - \dfrac{2}{\pi} \left(-\dfrac{\pi}{2} \ln(2) \right) \\ 2I &= \ln(\pi) + \ln(2) \\ I &= \dfrac{1}{2} \ln(2 \pi) \end{aligned}

In the above solution I used the result that $\int_{0}^{\pi/2} \ln(\sin(x))dx = -\dfrac{\pi}{2} \ln(2)$. This is proved here as follows:

\begin{aligned} J &=\int_{0}^{\pi/2} \ln(\sin(x))dx = \int_{0}^{\pi/2} \ln(\sin(\pi/2 - x))dx = \int_{0}^{\pi/2} \ln(\cos(x))dx \\ 2J &= \int_{0}^{\pi/2} \ln(\sin(x))dx + \int_{0}^{\pi/2} \ln(\cos(x))dx\\ 2J &= \int_{0}^{\pi/2} \ln(\sin(x) \cos(x))dx \\ 2J &= \int_{0}^{\pi/2} \ln(\sin(2x))dx - \int_{0}^{\pi/2} \ln(2)dx \\ 2J &= \dfrac{1}{2} \int_{0}^{\pi} \ln(\sin(x))dx - \dfrac{\pi}{2} \ln(2) \\ 2J &= \dfrac{1}{2}\left( 2\int_{0}^{\pi/2} \ln(\sin(x))dx \right) - \dfrac{\pi}{2} \ln(2) \\ 2J &= J - \dfrac{\pi}{2} \ln(2) \\ J &=- \dfrac{\pi}{2} \ln(2) \end{aligned}

- 4 years, 2 months ago

Problem 48:

Evaluate $\int _{ 0 }^{ \pi /2 }{ \frac { \csc ( x ) \sqrt { \tan x } }{ \sin x+\cos x } \, dx . }$

This problem has been solved by Surya Prakesh.

- 4 years, 2 months ago

Take the substitution $t= \tan x$.

\begin{aligned} \int_{0}^{\pi/2} \dfrac{ \csc x \sqrt{\tan x}}{\sin x + \cos x} dx &= 2\int_{0}^{\pi/2} \dfrac{\sqrt{\tan x}}{(1+\tan x ) (\sin 2x)} dx \\ &= 2\int_{0}^{\infty} \dfrac{\sqrt{t}}{(1+t)(2t)} dt \\ &= \int_{0}^{\infty} \dfrac{\sqrt{t}}{(1+t)(t)} dt \end{aligned}

Take $t = tan^2 (y)$.

$\int_{0}^{\infty} \dfrac{\sqrt{t}}{(1+t)(t)} dt = \int_{0}^{\pi /2} \dfrac{\tan y}{\sec^2 (y) \tan^2 (y)} (2 \tan(y) \sec^2 (y) dy) = 2 \int_{0}^{\pi/2}dy = \pi.$

- 4 years, 2 months ago

Problem 47:

Evaluate $\displaystyle \large \int _{ 0 }^{ \infty }{ { e }^{ -{ x }^{ 2 } }\ln { (x) } \, dx. }$

This problem has been solved by Julian Poon.

- 4 years, 2 months ago

Lemma $\int _{ 0 }^{ \infty }{ x^{ a }\cdot e^{ -x^{ 2 } }dx } =\frac { \Gamma \left( \frac { a+1 }{ 2 } \right) }{ 2 }$

Giving the substitution $x^2=t$, the integral becomes:

$\int _{ 0 }^{ \infty }{ x^{ a }\cdot e^{ -x^{ 2 } }dx } =\frac { 1 }{ 2 } \int _{ 0 }^{ \infty }{ t^{ \frac { a-1 }{ 2 } }\cdot e^{ -t }dt } =\frac { \Gamma \left( \frac { a+1 }{ 2 } \right) }{ 2 }$

So the required integral is

$=\frac { 1 }{ 4 } \left[ \Gamma \left( \frac { 1 }{ 2 } \right) \psi \left( \frac { 1 }{ 2 } \right) \right] =-\frac { 1 }{ 4 } \sqrt { \pi } (\gamma +\ln{(4)})$

To show that $\psi(1/2)=-\gamma-\ln(4)$, I'll use $\psi(a)=H_{a-1}-\gamma$

$\psi(1/2)=H_{-1/2} - \gamma$

From the definition of harmonic numbers:

$H_{-1/2}=H_{1/2} - 2$

$H_{1/2}=\int _{ 0 }^{ 1 }{ \frac { 1-\sqrt { x } }{ 1-x } dx } =2\int _{ 0 }^{ 1 }{ t\left( \frac { 1-t }{ 1-t^{ 2 } } \right) dt }$

I used the substitution $t=\sqrt{x}$ above

$\int _{ 0 }^{ 1 }{ t\left( \frac { 1-t }{ 1-t^{ 2 } } \right) dt } =\int _{ 0 }^{ 1 }{ \frac{t}{1+t}dt } =\int _{ 0 }^{ 1 }{ 1-\frac{1}{1+t} dt}=1-\ln(2)$

Hence

$H_{-1/2}=H_{1/2} - 2=2\left(1-\ln \left(2\right)\right)-2=-\ln(4)$

$\psi(1/2)=-\ln(4) - \gamma$

- 4 years, 2 months ago

Problem 46:

Evaluate

$\large \lim_{n \to \infty} \int_{0}^{{\pi} / {4}} n \left(\cos^{2n} x - \sin^{2n} x \right) \tan 2x \ \mathrm{d}x.$

- 4 years, 2 months ago

Take $\displaystyle f(n) = \int _{ 0 }^{ \frac { \pi }{ 4 } }{ n(\cos ^{ 2n }{ \theta } -\sin ^{ 2n }{ \theta } )\tan { (2\theta ) } d\theta }$

Take $cos(2\theta)=x$, to get :

$\displaystyle f(n) = \frac { n }{ { 2 }^{ n+1 } } \int _{ 0 }^{ 1 }{ \frac { { (1+x) }^{ n }-{ (1-x) }^{ n } }{ x } dx }$

We have to find $\displaystyle \lim _{ n\rightarrow \infty }{ f(n) }$

We will first find the functional equation :

$\displaystyle \frac { n }{ { 2 }^{ n+1 } } \int _{ 0 }^{ 1 }{ \frac { { (1+x) }^{ n }-{ (1-x) }^{ n } }{ x } dx } =\frac { n }{ { 2 }^{ n+1 } } \int _{ 0 }^{ 1 }{ \frac { { (1+x) }^{ n-1 }-{ (1-x) }^{ n-1 } }{ x } dx } +\frac { n }{ { 2 }^{ n+1 } } \int _{ 0 }^{ 1 }{ { (1+x) }^{ n-1 }+{ (1-x) }^{ n-1 }dx }$

$\displaystyle f(n) = \frac { nf(n-1) }{ 2(n-1) } +\frac { 1 }{ 2 }$

Let the limit be $L$

$\displaystyle \lim _{ n\rightarrow \infty }{ f(n) } =\lim _{ n\rightarrow \infty }{ \frac { nf(n-1) }{ 2(n-1) } +\frac { 1 }{ 2 } }$

Hence $\displaystyle L = \dfrac{1}{2}L+\dfrac{1}{2}$

$\Rightarrow \boxed{L=1}$

- 4 years, 2 months ago

Problem 45:

Prove that $\large \int_0^{\pi /2} \sin^6 (x) \tan (x ) \sin(\tan x) \, dx = \dfrac{5\pi}{96e} .$

This problem has been solved by Ishan Singh.

- 4 years, 2 months ago

Substitute $\displaystyle \tan x \mapsto x$

$\displaystyle \implies \text{I} = \int_{0}^{\infty} \dfrac{x^{7}\sin x}{(1+x^2)^4} \mathrm{d}x$

From solution of Problem 28, we have,

Differentiating $(*)$ $6$ times w.r.t. $m$ , $1$ time w.r.t. $a$, at $a=m=1$ and simplifying, we get,

$\displaystyle \text{I} = \boxed{\dfrac{5 \pi}{96e}}$

- 4 years, 2 months ago

Can you guys join Slack? I intend to form a discussion group for the series+integrals enthusiast for us.

Do let me know if you do not know how to log in.

- 4 years, 2 months ago

Hellolo sir, could I be part of this discussion? i'm also pretty interested in Integrals and series as well

- 3 years, 11 months ago

Join Slack first.

- 3 years, 11 months ago

- 3 years, 11 months ago

Thanks!

- 3 years, 11 months ago

How do you do you that? It isn't very clear

- 3 years, 11 months ago

I'm in.

- 4 years, 2 months ago

message me (3.14159han)

- 4 years, 2 months ago

- 3 years, 8 months ago

Hi participants. Contour integration is allowed from now onwards.

- 4 years, 2 months ago

Problem 44:

Evaluate $\displaystyle \int _{ 0 }^{ \infty }{ \left( \dfrac{ \ln { (1+x) } }{ { \ln }^{ 2 }(x)+{ \pi }^{ 2 } }\right) \frac { dx }{ {x}^{2} } } .$

Due to time constraint, the author decided to post their own solution.

- 4 years, 2 months ago

Lemma : $\displaystyle I = \int _{ 0 }^{ \infty }{ \frac { { e }^{ -ky } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy } =\frac{{ (-1) }^{ k }}{\pi} \int _{ k\pi }^{ \infty }{ \frac { sin(x) }{ x } dx }$ , for $k$ is a whole number.

Proof : We being with noting that :

$\displaystyle \frac { 1 }{ w } \int _{ 0 }^{ \infty }{ { e }^{ -st }sinwtdt } =\frac { 1 }{ { s }^{ 2 }+{ w }^{ 2 } }$

So we can rewrite the integral on the left side as :

$\displaystyle I = \frac { 1 }{ \pi } \int _{ 0 }^{ \infty }{ { e }^{ -ky }dy } \int _{ 0 }^{ \infty }{ { e }^{ -ty }sin\pi tdt } =\frac { 1 }{ \pi } \int _{ 0 }^{ \infty }{ sin\pi t\int _{ 0 }^{ \infty }{ { e }^{ -(k+t)y }dy } dt }$

$\displaystyle \Rightarrow I = \frac { 1 }{ \pi } \int _{ 0 }^{ \infty }{ \frac { sin\pi t }{ t+k } dt }$

$\displaystyle \Rightarrow I = \frac { 1 }{ \pi } \int _{ k }^{ \infty }{ \frac { sin(\pi t-\pi k) }{ t } dt }$

$\displaystyle \Rightarrow I = { (-1) }^{ k }\int _{ k }^{ \infty }{ \frac { sin(\pi t) }{ t } dt }$

$\displaystyle \Rightarrow I = { (-1) }^{ k }\int _{ k\pi }^{ \infty }{ \frac { sin(t) }{ t } dt }$

Hence proved

Solution :

We start with $x={e}^{-y}$ to get out integral as :

$\displaystyle I = \int _{ -\infty }^{ \infty }{ \frac { { e }^{ y }\ln { (1+{ e }^{ -y }) } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy }$

Break in into two parts :

$\displaystyle I = \int _{ 0 }^{ \infty }{ \frac { { e }^{ y }\ln { (1+{ e }^{ -y }) } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy } +\int _{ -\infty }^{ 0 }{ \frac { { e }^{ y }\ln { (1+{ e }^{ -y }) } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy }$

In the second part put $y=-x$ to get :

$\displaystyle I = \int _{ 0 }^{ \infty }{ \frac { { e }^{ y }\ln { (1+{ e }^{ -y }) } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy } +\int _{ 0 }^{ \infty }{ \frac { { e }^{ -x }\ln { (1+{ e }^{ x }) } }{ { x }^{ 2 }+{ \pi }^{ 2 } } dy }$

Manipulating it we have :

$\displaystyle I = \int _{ 0 }^{ \infty }{ \frac { { e }^{ y }\ln { (1+{ e }^{ -y }) } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy } +\int _{ 0 }^{ \infty }{ \frac { { e }^{ -y }\ln { (1+{ e }^{ -y }) } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy } +\int _{ 0 }^{ \infty }{ \frac { { ye }^{ -y } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy }$

$\displaystyle \Rightarrow I= \int _{ 0 }^{ \infty }{ \frac { ({ e }^{ y }+{ e }^{ -y })\ln { (1+{ e }^{ -y }) } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy } +\int _{ 0 }^{ \infty }{ \frac { { ye }^{ -y } }{ { y }^{ 2 }+{ \pi }^{ 2 } } dy }$

We write $I = J+K$

$\displaystyle J = \int _{ 0 }^{ \infty }{ \frac { ({ e }^{ x }+{ e }^{ -x })\ln { (1+{ e }^{ -x }) } }{ { x }^{ 2 }+{ \pi }^{ 2 } } } dx$

Wrting $\ln(1+{e}^{-x})$ in it's taylor series we have :

$\displaystyle J = \int _{ 0 }^{ \infty }{ \frac { ({ e }^{ x }+{ e }^{ -x }) }{ { x }^{ 2 }+{ \pi }^{ 2 } } \sum _{ r=1 }^{ \infty }{ \frac { { (-1) }^{ r-1 } }{ r } { e }^{ -rx } } } dx$

We interchange the integral and sum :

$\displaystyle J = \sum _{ r=1 }^{ \infty }{ \frac { { (-1) }^{ r-1 } }{ r } (\int _{ 0 }^{ \infty }{ \frac { { e }^{ -(r-1)x } }{ { x }^{ 2 }+{ \pi }^{ 2 } } } dx+\int _{ 0 }^{ \infty }{ \frac { { e }^{ -(r+1)x } }{ { x }^{ 2 }+{ \pi }^{ 2 } } } dx } )$

Using the lemma we have :

$\displaystyle J = \frac { 1 }{ \pi } \sum _{ r=1 }^{ \infty }{ \frac { 1 }{ r } (\int _{ (r-1)\pi }^{ \infty }{ \frac { \sin { (x) } }{ x } dx } +\int _{ (r+1)\pi }^{ \infty }{ \frac { \sin { (x) } }{ x } } dx) }$

Now it is worthy noting that :

$\displaystyle \int _{ r\pi }^{ \infty }{ \frac { \sin { (x) } }{ x } dx } =\int _{ 1 }^{ \infty }{ \frac { sin(r\pi x) }{ x } dx }$ , for all positive $r$, what if it was applied for $r=0$, then the left integral is $\frac{\pi}{2}$, while the write one is 0, so for $r=0$, we can write it as :

$\displaystyle \int _{ r\pi }^{ \infty }{ \frac { \sin { (x) } }{ x } dx } =\int _{ 1 }^{ \infty }{ \frac { sin(r\pi x) }{ x } dx } +\frac { \pi }{ 2 }$, for $r=0$, having said that we proceed further :

$\displaystyle J = \frac { 1 }{ \pi } \sum _{ r=1 }^{ \infty }{ \frac { 1 }{ r } (\int _{ \frac{1}{2} }^{ \infty }{ \frac { \sin { (2(r-1)\pi x) } +\sin { (2(r+1)\pi x) } }{ x } dx } ) } +\frac { 1 }{ 2 }$

$\displaystyle \Rightarrow J = \frac { 2 }{ \pi } \sum _{ r=1 }^{ \infty }{ \frac { 1 }{ r } (\int _{ \frac{1}{2} }^{ \infty }{ \frac { cos(2\pi x)\sin { (2r\pi x) } }{ x } dx } ) } +\frac { 1 }{ 2 }$

Again changing sum and integral we have :

$\displaystyle \Rightarrow J = \frac { 2 }{ \pi } \int _{ \frac { 1 }{ 2 } }^{ \infty }{ \cos { (2\pi x) } \sum _{ r=1 }^{ \infty }{ \frac { \sin { (2r\pi x) } }{ rx } } dx } +\frac { 1 }{ 2 }$

It is a well known result that :

$\displaystyle \sum _{ r=1 }^{ \infty }{ \frac { \sin { (2r\pi x) } }{ r\pi } } =\left\lfloor x \right\rfloor +\frac { 1 }{ 2 }-x$

Using this we have :

$\displaystyle J = 2\int _{ \frac { 1 }{ 2 } }^{ \infty }{ \frac { \cos { (2\pi x) } }{ x } (-x+\left\lfloor x \right\rfloor +\frac { 1 }{ 2 } )dx } +\frac { 1 }{ 2 }$

Leaving this let's evaluate $K$ first :

Using the same techniques as used in proving the lemma, we can show that :

$\displaystyle K = \int _{ 0 }^{ \infty }{ \frac { x{ e }^{ -x } }{ { x }^{ 2 }+{ \pi }^{ 2 } } dx } =-\int _{ 1/2 }^{ \infty }{ \frac { \cos { (2\pi x) } }{ x } dx }$

Now $\displaystyle I = J+K = 2\int _{ \frac { 1 }{ 2 } }^{ \infty }{ \frac { \cos { (2\pi x) } }{ x } (-x+\left\lfloor x \right\rfloor +\frac { 1 }{ 2 } )dx } +\frac { 1 }{ 2 } -\int _{ 1/2 }^{ \infty }{ \frac { \cos { (2\pi x) } }{ x } dx } =\frac { 1 }{ 2 } +2\int _{ 1/2 }^{ \infty }{ \frac { \cos { (2\pi x) } }{ x } \left\{ x \right\} dx }$

$\displaystyle I = \frac { 1 }{ 2 } +2\int _{ 1 }^{ \infty }{ \frac { \cos { (2\pi x) } }{ x } \left\{ x \right\} dx } +2\int _{ \frac { 1 }{ 2 } }^{ 1 }{ \cos { (2\pi x) } dx } =\frac { 1 }{ 2 } +2\int _{ 1 }^{ \infty }{ \frac { \cos { (2\pi x) } }{ x } \left\{ x \right\} dx }$

Now we will be evaluating :

$\displaystyle M = \int _{ 1 }^{ \infty }{ \frac { \cos { (2\pi x) } }{ x } \left\{ x \right\} dx }$

It can be written as :

$\displaystyle M = \sum _{ n=1 }^{ \infty }{ \int _{ n }^{ n+1 }{ \frac { \cos { (2\pi x) } }{ x } (x-n)dx } }$

$\displaystyle \Rightarrow M = -\sum _{ n=1 }^{ \infty }{ n\int _{ n }^{ n+1 }{ \frac { \cos { (2\pi x) } }{ x } dx } }$

$M$ can also be written as :

$\displaystyle M =-\sum _{ n=1 }^{ \infty }{ \int _{ n }^{ \infty }{ \frac { \cos { (2\pi x) } }{ x } dx } }$

This manipulation can easily be justified with the help of properties of double summations :

Using integration by parts we have :

$\displaystyle M = \sum _{ n=1 }^{ \infty }{ \int _{ n }^{ \infty }{ \frac { \sin { (2\pi x) } }{ 2\pi { x }^{ 2 } } dx } } -\sum _{ n=1 }^{ \infty }{ \frac { \sin { (2\pi n) } }{ 2\pi n } }$

$\displaystyle \Rightarrow M = \sum _{ n=1 }^{ \infty }{ \int _{ n }^{ \infty }{ \frac { \sin { (2\pi x) } }{ 2\pi { x }^{ 2 } } dx } }$

$\displaystyle \Rightarrow M = \sum _{ n=1 }^{ \infty }{ \int _{ 1 }^{ \infty }{ \frac { \sin { (2\pi nx) } }{ 2\pi { nx }^{ 2 } } dx } }$

Again changing sum and integral we have :

$\displaystyle \Rightarrow M = \quad \int _{ 1 }^{ \infty }{ \sum _{ n=1 }^{ \infty }{ \frac { \sin { (2\pi nx) } }{ 2\pi { nx }^{ 2 } } } dx }$

Using the result we have used above we have :

$\displaystyle M = \frac { 1 }{ 2 } \int _{ 1 }^{ \infty }{ \frac { 1 }{ 2{ x }^{ 2 } } -\frac { \left\{ x \right\} }{ { x }^{ 2 } } dx }$

$\displaystyle \Rightarrow M = \quad \frac { 1 }{ 2 } \int _{ 1 }^{ \infty }{ \frac { 1 }{ 2{ x }^{ 2 } } -\frac { \left\{ x \right\} }{ { x }^{ 2 } } dx }$

For solving $\displaystyle H = \int _{ 1 }^{ \infty }{ \frac { \left\{ x \right\} }{ { x }^{ 2 } } dx }$

write it as a sum :

$\displaystyle H = \sum _{ n=1 }^{ \infty }{ \int _{ n }^{ n+1 }{ \frac { 1 }{ x } -\frac { n }{ { x }^{ 2 } } dx } }$

$\displaystyle \Rightarrow H = \sum _{ n=1 }^{ \infty }{ \ln { (n+1) } -\ln { (n) } -\frac { 1 }{ n+1 } }$

$\displaystyle H = \lim _{ n\rightarrow \infty }{ 1+\ln { (n+1)-{ H }_{ n+1 } } } =1-\gamma$

Hence $\displaystyle M = \frac { \gamma }{ 2 } -\frac { 1 }{ 4 }$

Finally we got :

$\Large \boxed { I=\gamma }$

- 4 years, 2 months ago

Problem 43:

Evaluate $\large \int \dfrac{x^3 e^{x^2} }{(x^2 + 1)^2} \, dx .$

This problem has been solved by Ronak Agarwal.

- 4 years, 2 months ago

Put ${x}^{2}=y$ to get the integral as :

$\displaystyle I = \frac { 1 }{ 2 } \int { \frac { y{ e }^{ y } }{ { (y+1) }^{ 2 } } dy }$

It can be also written as :

$\displaystyle I = \frac { 1 }{ 2 } \int { \frac { { e }^{ y } }{ (y+1) } -\frac { { e }^{ y } }{ { (y+1) }^{ 2 } } dy } =\frac { 1 }{ 2 } \int { \frac { d(\frac { { e }^{ y } }{ y+1 } ) }{ dy } dy }$

$I = \dfrac { { e }^{ y } }{ 2(y+1) } +C=\dfrac { { e }^{ { x }^{ 2 } } }{ 2({ x }^{ 2 }+1) } +C$

- 4 years, 2 months ago

Another way to solve is if we consider quotient rule in reverse its easy to guess our integral to be of the form ce^{x^2}/[x^2+1] it remains to solve for c and we're done

- 1 year, 9 months ago

Problem 42:

Evaluate $\displaystyle \large \int _{ 0 }^{ \frac{\pi}{2} }{ \ln { \left( 2\cos \left( \frac { x }{ 2 } \right) \right) } \,dx }$

This problem has been solved by Surya Prakesh.

- 4 years, 2 months ago

$\ln \left( 2\cos \left(\dfrac{x}{2} \right) \right) = \ln \left(e^{ix/2} + e^{-ix/2} \right) = \ln \left( 1 + e^{ix} \right) - i\dfrac{x}{2}$

So,

\begin{aligned} \int_{0}^{\pi / 2} \ln \left( 2\cos \left(\dfrac{x}{2} \right) \right) dx &= \int_{0}^{\pi / 2} \ln \left( 1 + e^{ix} \right) dx - i \int_{0}^{\pi / 2}\dfrac{x}{2} dx \\ &= \int_{0}^{\pi /2} \sum_{r=1}^{\infty} \dfrac{(-1)^{r+1} e^{irx}}{r} dx - i \dfrac{\pi^2}{16} \end{aligned}

We shall evaluate the first integral

\begin{aligned} \int_{0}^{\pi /2} \sum_{r=1}^{\infty} \dfrac{(-1)^{r+1} e^{irx}}{r} dx &= \sum_{r=1}^{\infty} \dfrac{(-1)^{r+1}}{r} \int_{0}^{\pi /2} e^{irx} dx \\ &= \sum_{r=1}^{\infty} \dfrac{(-1)^{r+1}}{r} \left[ \dfrac{e^{irx}}{ir} \right|_{0}^{\pi/2}\\ &= \dfrac{1}{i} \sum_{r=1}^{\infty} \dfrac{(-1)^{r+1} \left(e^{ir \pi /2} - 1\right)}{r^2} \end{aligned}

Now,

$e^{ ir\pi /2 }-1=\quad \begin{cases} i-1 & \quad \quad r=4k+1 \\ -2 & \quad \quad r=4k+2 \\ -i-1 & \quad \quad r=4k+3 \\ 0 & \quad \quad r=4k+4 \end{cases}$

\begin{aligned}\sum_{r=1}^{\infty} \dfrac{(-1)^{r+1} \left(e^{ir \pi /2} - 1\right)}{r^2} &= \sum_{k=0}^{\infty} \dfrac{(-1)^{4k+2}(i-1)}{(4k+1)^{2}} + \sum_{k=0}^{\infty} \dfrac{(-1)^{4k+3}(-2)}{(4k+2)^{2}} + \sum_{k=0}^{\infty} \dfrac{(-1)^{4k+4}(-i-1)}{(4k+3)^{2}} + \sum_{k=0}^{\infty} \dfrac{(-1)^{4k+5}(0)}{(4k+4)^{2}} \\ &= \sum_{k=0}^{\infty} \dfrac{(i-1)}{(4k+1)^{2}} + \sum_{k=0}^{\infty} \dfrac{2}{(4k+2)^{2}} + \sum_{k=0}^{\infty} \dfrac{-i-1}{(4k+3)^{2}} + 0 \\ &= \sum_{k=0}^{\infty} \dfrac{(i-1)}{(4k+1)^{2}} + \sum_{k=0}^{\infty} \dfrac{2}{(4k+2)^{2}} + \sum_{k=0}^{\infty} \dfrac{-i-1}{(4k+3)^{2}} \end{aligned}

$\sum_{k=0}^{\infty} \dfrac{1}{(4k+1)^2} = \dfrac{1}{16} \zeta \left(2 , \dfrac{1}{4} \right) \\ \sum_{k=0}^{\infty} \dfrac{1}{(4k+2)^2} = \dfrac{1}{16} \zeta \left(2 , \dfrac{1}{2} \right) \\ \sum_{k=0}^{\infty} \dfrac{1}{(4k+3)^2} = \dfrac{1}{16} \zeta \left(2 , \dfrac{3}{4} \right)$

So,

\begin{aligned} \sum_{r=1}^{\infty} \dfrac{(-1)^{r+1} \left(e^{ir \pi /2} - 1\right)}{r^2} &= (i-1) \dfrac{1}{16} \zeta \left(2 , \dfrac{1}{4} \right) + 2 \dfrac{1}{16} \zeta \left(2 , \dfrac{1}{2} \right) + (-i-1) \dfrac{1}{16} \zeta \left(2 , \dfrac{3}{4} \right)\\ &= \dfrac{1}{16} \left( i \left( \zeta \left(2 , \dfrac{1}{4} \right) - \zeta \left(2 , \dfrac{3}{4} \right) \right) - \left( \zeta \left(2 , \dfrac{1}{4} \right) + \zeta \left(2 , \dfrac{3}{4} \right) \right) + 2 \zeta \left(2 , \dfrac{1}{2} \right) \right) \end{aligned}

1:
$\zeta \left(2 , \dfrac{1}{4} \right) - \zeta \left(2 , \dfrac{3}{4} \right) = 16 \sum_{k=0}^{\infty} \left(\dfrac{1}{(4k+1)^2} - \dfrac{1}{(4k+3)^2}\right) = 16 \sum_{k=0}^{\infty} \dfrac{(-1)^{k}}{(2k+1)^2} = 16G$

2:

\begin{aligned} \zeta \left(2 , \dfrac{1}{4} \right) + \zeta \left(2 , \dfrac{3}{4} \right) &= 16 \sum_{k=0}^{\infty} \left(\dfrac{1}{(4k+1)^2} + \dfrac{1}{(4k+3)^2}\right) \\ &= 16 \sum_{k=0}^{\infty} \dfrac{1}{(2k+1)^2}\\ &=16 \dfrac{\pi^2}{8}\\ &= 2 \pi^2 \end{aligned}

3:

$\zeta \left(2 , \dfrac{1}{2} \right) = 4 \sum_{k=0}^{\infty} \dfrac{1}{(2k+1)^2} = 4 \dfrac{\pi^2}{8} = \dfrac{\pi^2}{2}$

So,

\begin{aligned} \sum_{r=1}^{\infty} \dfrac{(-1)^{r+1} \left(e^{ir \pi /2} - 1\right)}{r^2} &= \dfrac{1}{16} \left( i \left( \zeta \left(2 , \dfrac{1}{4} \right) - \zeta \left(2 , \dfrac{3}{4} \right) \right) - \left( \zeta \left(2 , \dfrac{1}{4} \right) + \zeta \left(2 , \dfrac{3}{4} \right) \right) + 2 \zeta \left(2 , \dfrac{1}{2} \right) \right)\\ &= \dfrac{1}{16} \left( i (16G) - (2\pi^2) + 2 \dfrac{\pi^2}{2} \right)\\ &= iG - \dfrac{\pi^2}{16} \end{aligned}