Mathematics

# Evaluate the following integral:$\int { \cfrac { \sin { x } }{ { \left( 1+\cos { x } \right) }^{ 2 } } } dx$

##### SOLUTION
Let
$t=1+\cos{x}\Rightarrow\,dt=-\sin{x}\,dx$

$\Rightarrow\,-dt=\sin{x}\,dx$

$\displaystyle\int{\dfrac{\sin{x}dx}{{\left(1+\cos{x}\right)}^{2}}}$

$=-\displaystyle\int{\dfrac{dt}{{t}^{2}}}$

$=-\displaystyle\int{{t}^{-2}\,dt}$

$=-\dfrac{{t}^{-2+1}}{-2+1}+c$

$=\dfrac{1}{t}+c$

$=\dfrac{1}{1+\cos{x}}+c$ where $t=1+\cos{x}$

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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
Evaluate: $\displaystyle \int_{0}^{1} \ x+x^2 dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\displaystyle \int \frac{dx}{(x + 2) (x^{2} + 1)} = a\ln (1 + x^{2}) + b\tan^{-1} x + \frac{1}{5} \ln | x + 2 | + C$ then
• A. $a = - \displaystyle \frac{1}{10}$, $b = - \displaystyle \frac{2}{5}$
• B. $a = \displaystyle \frac{1}{10}$, $b = \displaystyle \frac{2}{5}$
• C. $a = \displaystyle \frac{1}{10}$, $b = - \displaystyle \frac{2}{5}$
• D. $a = - \displaystyle \frac{1}{10}$, $b = \displaystyle \frac{2}{5}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
lf $I_{m,n}=\displaystyle \int\frac{x^{m}}{(\log x)^{n}}dx$ then
$(m+1)I_{m,n}-n.I_{m,n+1}=$
• A. $\displaystyle \frac{x^{m}}{(\log x)^{n}}+c$
• B. $-\displaystyle \frac{x^{m}}{(\log x)^{n}}+c$
• C. $-\displaystyle \frac{x^{m+1}}{(\log x)^{n}}+c$
• D. $\displaystyle \frac{x^{m+1}}{(\log x)^{n}}+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
The value of integral  $\displaystyle \int e^{x}\left ( \frac{1}{\sqrt{1+x^{2}}}+\frac{1-2x^{2}}{\sqrt{(1+x^{2})^{5}}} \right )dx$ is equal to
• A. $\displaystyle e^{x}\left ( \frac{1}{\sqrt{1+x^{2}}}-\frac{x}{\sqrt{(1+x^{2})^{3}}} \right )+c$
• B. $\displaystyle e^{x}\left ( \frac{1}{\sqrt{1+x^{2}}}+\frac{x}{\sqrt{(1+x^{2})^{5}}} \right )+c$
• C. none of these
• D. $\displaystyle e^{x}\left ( \frac{1}{\sqrt{1+x^{2}}}+\frac{x}{\sqrt{(1+x^{2})^{3}}} \right )+c$

$\underset {n\rightarrow \infty}{lim}\dfrac{1^2+2^2+3^2+.....+n^2}{n^3}=.................$