Mathematics

# Evaluate the following integral:$\int { \cfrac { 1+\cos { x } }{ { \left( x+\sin { x } \right) }^{ 3 } } } dx\quad$

##### SOLUTION
Let
$t=x+\sin{x}\Rightarrow\,dt=\left(1+\cos{x}\right)dx$

$\displaystyle\int{\dfrac{\left(1+\cos{x}\right)dx}{{\left(x+\sin{x}\right)}^{3}}}$

$=\displaystyle\int{\dfrac{dt}{{t}^{3}}}$

$=\displaystyle\int{{t}^{-3}dt}$

$=\dfrac{{t}^{-3+1}}{-3+1}+c$

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

$=\dfrac{-1}{2{\left(x+\sin{x}\right)}^{2}}+c$ where $t=x+\sin{x}$

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

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle \int_{1}^{2}\mathrm{x}^{2\mathrm{x}}[1+\log \mathrm{x}]\mathrm{d}\mathrm{x}=$
• A. $\displaystyle \frac{9}{2}$
• B. $\displaystyle \frac{11}{2}$
• C. $\displaystyle \frac{13}{2}$
• D. $\displaystyle \frac{15}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int \frac{x^{4}-2}{x^{2} \sqrt{x^{4}+x^{2}+2}} d x$
• A. $\frac{\sqrt{x^{4}+x^{2}+2}}{|x|}+c$
• B. $\frac{\sqrt{x^{4}+1}}{|x|}+c$
• C. $\frac{\sqrt{x^{4}+2}}{|x|}+c$
• D. $\frac{\sqrt{x^{4}+x^{2}+1}}{|x|}+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\displaystyle \int { \dfrac { { x }^{ 2 }-5x-1 }{ { x }^{ 4 }+{ x }^{ 2 }+1 } dx }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following definite integral:

$\displaystyle \int _{0}^1 \dfrac {1-x}{1+x}dx$

Integrate $\displaystyle\int {\sqrt {\frac{{1 + x}}{{1 - x}}} dx\ , on ( - 1,1)\,.}$