Mathematics

# Evaluate:$\displaystyle \int_{0}^{\pi} \dfrac{x}{1 + sinx}$

##### SOLUTION
Let $I = \displaystyle \int_{0}^{\pi} \dfrac{x}{1 + sin x}dx$ ....(i)
and $I = \displaystyle \int_{0}^{\pi} \dfrac{\pi - x}{1 + sin(\pi - x)}dx = \displaystyle \int_{0}^{\pi} \dfrac{\pi - x}{1 + sin x}dx$ .....(ii)

On adding Eqs. (i) and (ii), we get
$2I = \pi \displaystyle \int_{0}^{\pi} \dfrac{1}{1 + sin x}dx$
$= \pi \displaystyle \int_{0}^{\pi} \dfrac{(1 - sin \,x)dx}{(1 + sin \,x)(1 - sin\,x)}$
$= \pi \displaystyle \int_{0}^{\pi} \dfrac{(1 - sin \,x)dx}{cos^2x}$
$= \pi \displaystyle \int_{0}^{\pi} (sec^2 x - tan \,x . sec\,x)dx$
$= \pi \displaystyle \int_{0}^{\pi} sec^2 xdx - \pi \displaystyle \int_{0}^{\pi} sec \,x \,tan \,x \,dx$
$= \pi[ (tan\,x)_{0}^{\pi} - (sec \,x)_{0}^{\pi}]$
$= \pi (tan \,pi - tan \,0 - sec \pi + sec \,0)$
$\Rightarrow 2I = \pi (0 - 0 + 1 + 1) = 2 \,\pi$
$2I = 2 \pi$
$\therefore I = \pi$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Single Correct Hard
If $\displaystyle I_{t}=\int_{0}^{\dfrac{\pi }{2}}\frac{\sin^{2}tx}{\sin^{2}x}dx$ then ,$I_{1},I_{2},I_{3}$ are in
• A. H.P.
• B. G.P.
• C. None of these
• D. A.P.

Asked in: Mathematics - Arithmetic Progressions

1 Verified Answer | Published on 17th 08, 2020

Q2 Subjective Medium
$\int {{e^x}(\cos x - \sin x)dx}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\displaystyle \int { { e }^{ x }\cos ^{ 2 }{ x } dx }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Evaluate : $\displaystyle\int^1_{-1} \log(x+\sqrt{x^2+1})dx$
• A. $\log \dfrac{1}{2}$
• B. $\log 2$
• C. $\dfrac{1}{2}\log 2$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q5 Assertion & Reason Hard
##### ASSERTION

If $n>1$ then Statement -1: $\displaystyle \int_{0}^{\infty}\frac{dx}{1+x^{n}}=\int_{0}^{1}\frac{dx}{(1-x^{n})^{1/n}}$

##### REASON

Statement -2: $\displaystyle \int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$

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