Mathematics

# Evaluate:$\displaystyle \int \dfrac { x ^ { 3 } } { ( x + 1 ) ^ { 2 } } d x$

##### SOLUTION
$I=\displaystyle\int \frac{x^{3}}{(x+1)^{2}}dx$

$I=\displaystyle\int \frac{x^{2}}{(x+1)^{2}}xdx$

$u= x+1$

$\Rightarrow x=u-1, dx=du$

$I=\displaystyle\int \frac{(u-1)^{2}(u-1)du}{u^{2}}$

$=\displaystyle\int \frac{(u-1)^{3}}{u^{2}}du$

$=\displaystyle\int \frac{u^{3}-3u^{2}+3u-1}{u^{2}}.du$

$=\displaystyle\int u-3+\frac{3}{u}-\frac{1}{u^{2}}du$

$=\dfrac{u^{2}}{2}-3u+3ln\left | u \right |+\dfrac{1}{u}+c$

$=\dfrac{(x+1)^{2}}{2}-3(x+1)+3ln\left | x+1 \right |+\dfrac{1}{x+1}+c$

Its FREE, you're just one step away

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

#### Realted Questions

Q1 Subjective Medium
Evaluate $\displaystyle\int_{0}^{2}x\sqrt{x+2} \ dx$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \dfrac{2x\log(1+x^2)}{1+x^2}dx$
• A. $\log(1+x^2)+c$
• B. $2\log(1+x^2)+c$
• C. none of these
• D. $\dfrac{[\log(1+x^2)]^2}{2}+c$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
$\int { \frac { \ln { \left( 1+{ e }^{ x } \right) } }{ { e }^{ x } } } dx$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 One Word Hard
If $I=\displaystyle \int_{0}^{1}\displaystyle \frac{dx}{\sqrt{x}+\sqrt{2-x}}$ then the value of $\left ( I+\log \left ( \sqrt{2}-1 \right ) \right )^{2}$ is

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Subjective Easy
Evaluate $\int \dfrac{e^x-e^{-x}}{e^x+e^{-x}}dx$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020