Mathematics

# Solve :$\displaystyle\int { \cfrac { { x }^{ 2 }+3x-1 }{ { \left( x+1 \right) }^{ 2 } } dx }$

##### SOLUTION
$\displaystyle\int{\dfrac{\left({x}^{2}+3x-1\right)dx}{{\left(x+1\right)}^{2}}}$

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

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

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

$=\displaystyle\int{\dfrac{{\left(x+1\right)}^{2}dx}{{\left(x+1\right)}^{2}}}+\displaystyle\int{\dfrac{\left(x+1-3\right)dx}{{\left(x+1\right)}^{2}}}$

$=\displaystyle\int{dx}+\displaystyle\int{\dfrac{\left(x+1\right)dx}{{\left(x+1\right)}^{2}}}-3\displaystyle\int{\dfrac{dx}{{\left(x+1\right)}^{2}}}$

$=\displaystyle\int{dx}+\displaystyle\int{\dfrac{dx}{\left(x+1\right)}}-3\displaystyle\int{\dfrac{dx}{{\left(x+1\right)}^{2}}}$

$=x+\log{\left|x+1\right|}+\dfrac{3}{x+1}+c$

Its FREE, you're just one step away

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

#### Realted Questions

Q1 Subjective Medium
Evaluate: $\displaystyle \int{e^x}\begin{pmatrix}\dfrac{x-1}{x^2}\end{pmatrix}dx$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Integrate $\dfrac {e^{\tan^{-1}}x}{1 + x^{2}}$ with respect to $x$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
If f(x) is a function satisfying $f\left( \dfrac{1}{x}\right)+x^2f(x)=0$ for all non-zero x, then $\displaystyle \int_{sin\Theta}^{cosec\Theta}f(x)dx$ is equal to

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If  $\displaystyle \int_{0}^{k}\frac{\cos x}{1+\sin^{2}x}dx=\frac{\pi}{4}$ then ${k}=?$
• A. $1$
• B. $\pi/4$
• C. $\pi/6$
• D. $\pi/2$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$