Mathematics

# $\int_{0}^{\pi/2}\dfrac {\sin x \cos x}{1+\sin^{4}x}dx$

##### SOLUTION
$\begin{array}{l} \int _{ }^{ }{ \frac { { \cos \left( x \right) \sin \left( x \right) } }{ { { { \sin }^{ 4 } }\left( x \right) +1 } } dx } \\ Substitute\, u={ \sin ^{ 2 } }x,dx=\frac { 1 }{ { 2\cos x\sin x } }du \\ =\frac { 1 }{ 2 } \int _{ }^{ }{ \frac { 1 }{ { 1+{ u^{ 2 } } } } du } \\ =\frac { 1 }{ 2 } { \tan ^{ -1 } }u+C \end{array}$

Apply the limit, we get
$\int_0^{\frac{\pi }{2}} {\frac{{\cos \left( x \right)\sin \left( x \right)}}{{{{\sin }^4}\left( x \right) + 1}}dx = \frac{\pi }{8}}$

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

#### Realted Questions

Q1 Subjective Medium
$\text { Evaluate: } \displaystyle \int_{-\pi / 2}^{\pi / 2} \dfrac{\cos x}{1+e^{x}} \mathrm{d} \mathrm{x}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Value of $\int \:e^x\:\frac{\left(x^2+3x+3\right)}{\left(x+2\right)^2}dx\:$ is
• A. $e^x\left[\dfrac{x-1}{x+2}\right]+c$
• B. $e^x\left[\dfrac{x+1}{x-2}\right]+c$
• C. None of these
• D. $e^x\left[\dfrac{x+1}{x+2}\right]+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 TRUE/FALSE Hard
If $f(a-x)=-f(x)$, then $\displaystyle \int_{0}^{a}f(x)dx=0$.

• A. False
• B. True

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If, $\displaystyle \int \dfrac{e^x - 1}{e^x + 1} dx = f(x) + c$, then $f(x)$ is equal to
• A. $2 \, \log (e^x + 1)$
• B. $\log (e^x - 1)$
• C. $\log (e^{2x} + 1)$
• D. $2 \, \log (e^x + 1) - x$

Evaluate : $\int{\dfrac{2x+2}{2x^{2}+4x-3}dx}$