Mathematics

Evaluate: $\int _ { 0 } ^ { \pi } \dfrac { x \sin x } { 1 + \cos ^ { 2 } x } d x$

SOLUTION
Let $I=\displaystyle\int^{\pi}_0\dfrac{x\sin x}{1+\cos^2x}dx$ ………$(1)$

$I=\displaystyle\int^{\pi}_0\dfrac{(\pi -x)\sin(\pi-x)}{1+\cos^2(\pi-x)}dx$

$=\displaystyle\int^{\pi}_0\dfrac{(\pi -x)\sin x}{1+\cos^2x}dx$ ……..$(2)$

Adding both $(1)$ and $(2)$, we get

$2I=\displaystyle\int^{\pi}_0\dfrac{\pi\sin x}{1+\cos^2x}dx$

Let $t=\cos x$             $\cos 0=1$

$dt=-\sin xdx$              $\cos\pi =-1$

So, $2I=\displaystyle\int^{-1}_1\dfrac{-\pi dt}{1+t^2}$

$2I=\pi\displaystyle\int^1_{-1}\dfrac{dt}{1+t^2}=\pi\left[\tan^{-1}t\right]^1_{-1}$

$2I=\pi\left[\dfrac{\pi}{4}-\left(-\dfrac{\pi}{4}\right)\right]$

$2I=\dfrac{\pi^2}{2}$

$I=\dfrac{\pi^2}{4}$

$I=\left(\dfrac{\pi}{2}\right)^2$.

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

Realted Questions

Q1 Subjective Medium
$\displaystyle \int^{\pi}_{0}\sqrt {1+\sin 2x }dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Assertion & Reason Medium
ASSERTION

$\displaystyle \int \left ( \frac{1}{1 + x^4} \right )dx = \tan^{-1} (x^2) + C$

REASON

$\displaystyle \int \frac{1}{1 + x^2} dx = \tan^{-1}x + C$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Find an anti derivative (or integral) of the given function by the method of inspection.
$\cos 3x$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Let $F\left( x \right) =f\left( x \right) +f\left( \dfrac { 1 }{ x } \right)$, where $\displaystyle f\left( x \right) =\int _{ 1 }^{ x }{ \dfrac { \log { t } }{ 1+t } } dt$. Then $F\left( e \right)$ equals -
• A. $0$
• B. $1$
• C. $2$
• D. $\dfrac { 1 }{ 2 }$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$