Mathematics

# The value of integral  $\int _ { \pi / 4 } ^ { 3 \pi / 4 } \dfrac { x } { 1 + \sin x } d x$  is :

$\pi ( \sqrt { 2 } - 1 )$

##### SOLUTION
$\\I=\int_{(\tfrac{\pi}{4})}^{(\tfrac{3\pi}{4})}(\dfrac{x}{1+sinx})dx$

$=\int_{(\tfrac{\pi}{4})}^{(\tfrac{3\pi}{4})}(\dfrac{\pi-x}{1+sinx})dx$

$I=\pi\int_{(\tfrac{\pi}{4})}^{(\tfrac{3\pi}{4})}(\dfrac{1}{1+sinx})dx-I\\or\>2I=\pi\int_{(\dfrac{\pi}{4})}^{(\dfrac{3\pi}{4})}(\dfrac{1-sinx}{1^2-sin^2x})dx$

$I=(\tfrac{\pi}{2})\int_{(\tfrac{\pi}{4})}^{(\tfrac{3\pi}{4})}\left((\dfrac{1}{cos^2x})-(\dfrac{sinx}{cos^2x})\right)dx$

$=(\dfrac{\pi}{2})\int_{(\tfrac{\pi}{4})}^{(\tfrac{3\pi}{4})}(sec^2x-tanx\>secx)dx$

$=(\dfrac{\pi}{2})\left[tanx-secx\right]_{\pi/4}^{3\pi/4}$

$=(\dfrac{\pi}{2})\left[\left(tan(\dfrac{3\pi}{4})-sec(\dfrac{3\pi}{4})\right)-\left(tan(\dfrac{\pi}{4})-sec(\dfrac{\pi}{4})\right)\right]$

$=(\dfrac{\pi}{2})[(-1+\sqrt{2})-(1-\sqrt{2})]\\=(\dfrac{\pi}{2})(2\sqrt{2})-2$

$=\pi(\sqrt{2}-1)$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Hard
$\int\dfrac{x ^2+37}{(x^2-7)(x^2+4)}dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
The value of $2\displaystyle \int \sin x \text{ cosec } 4x\ dx$ is equal to:
• A. $\dfrac{1}{2\sqrt2}\ln \left|\dfrac{1+\sqrt2\sin x}{1-\sqrt2\sin x}\right|- \dfrac{1}{2}\ln \left|\dfrac{1+\sin x}{\cos x}\right|+c$
• B. $\dfrac{1}{2\sqrt2}\ln \left|\dfrac{1-\sqrt2\sin x}{1+\sqrt2\sin x}\right|- \dfrac{1}{4}\ln \left|\dfrac{1+\sin x}{1-\sin x}\right|+c$
• C. $\dfrac{1}{2\sqrt2}\ln \left|\dfrac{1-\sqrt2\sin x}{1+\sqrt2\sin x}\right|+ \dfrac{1}{2}\ln \left|\dfrac{1+\sin x}{\cos x}\right|+c$
• D. $\dfrac{1}{2\sqrt2}\ln \left|\dfrac{1+\sqrt2\sin x}{1-\sqrt2\sin x}\right|- \dfrac{1}{4}\ln \left|\dfrac{1+\sin x}{1-\sin x}\right|+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int_{a + c}^{b + c} f(x) dx$ is equal to
• A. $\displaystyle \int_{a}^{b} f(x - c) dx$
• B. $\displaystyle \int_{a}^{b} f(x) dx$
• C. $\displaystyle \int_{a + x}^{b + x } f(x) dx$
• D. $\displaystyle \int_{a}^{b} f(x + c) dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate :
$\int {\frac{{x + 1}}{{{x^2} + 3x + 12}}} dx$

Let $\displaystyle I_{1}=\int_{0}^{1}(1-x^{2})^{1/3} dx$  &  $\displaystyle I_{2}=\int_{0}^{1}(1-x^{3})^{1/2} dx$