Mathematics

$\int$ $\dfrac {dx}{(x^{2}+1)(x+3)}$

SOLUTION
We have,
$I=\int \dfrac {1}{(x^{2}+1)(x+3)}$

Let
$\dfrac {1}{(x^{2}+1)(x+3)}=\dfrac{Ax+B}{x^2+1}+\dfrac{C}{x+3}$

$1=(Ax+B)(x+3)+C(x^2+1)$

$1=Ax^2+Bx+3Ax+3B+Cx^2+C$             $............(1)$

On solving equation $(1)$, we get
$A=-\dfrac{1}{10}, B=\dfrac{3}{10}, C=\dfrac{1}{10}$

Therefore,
$I=\displaystyle \int \dfrac{1}{10}\left(\dfrac{-x+3}{x^2+1}\right)+\dfrac{1}{10(x+3)}\ dx$

$I=- \dfrac{1}{10}\int \left(\dfrac{x}{x^2+1}\right)dx+\dfrac{3}{10}\int \dfrac{dx}{x^2+1}+\int \dfrac{1}{10(x+3)}\ dx$

$I=- \dfrac{1}{20}\int \left(\dfrac{2x}{x^2+1}\right)dx+\dfrac{3}{10}\int \dfrac{dx}{x^2+1}+\int \dfrac{1}{10(x+3)}\ dx$

$I=- \dfrac{1}{20}\ln(x^2+1)+\dfrac{3}{10}\tan^{-1}(x^2+1)+ \dfrac{1}{10}\ln (x+3)+C$

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

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int\frac{\cos x}{(1+\sin x)(2+\sin x)}dx=$
• A. $\displaystyle \log|\frac{2+\sin x}{1+\sin x}|+c$
• B. $\log|(1+sinx)(2+\sin x)|+c$
• C. $\log|(1+sinx)+(2+\sin x)|+c$
• D. $\displaystyle \log|\frac{1+\sin x}{2+\sin x}|+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Solve: $\int 3^x.e^x dx =$?

1 Verified Answer | Published on 17th 09, 2020

Q4 Multiple Correct Hard
If $\displaystyle \int xe^{-5x^2} \: sin \: 4x^2 dx = Ke^{-5x^2} (A \: sin \: 4x^2 + B \: cos \: 4x^2) + C$. Then
• A. $\dfrac {1}{82}, -5, 4$
• B. $\dfrac {1}{82}, \dfrac{-1}{5}, \dfrac{-1}{4}$
• C. $\dfrac {1}{82}, 5, 4$
• D. $\dfrac {1}{82}, -5, -4$

$\int(2x^2-3 \, sin x+5 \sqrt{x})dx$