Evaluate the following intregals: 5 (x^2+1)(x+2) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{5}{(\text{x}^2+1)(\text{x}+2)}\text{ dx}$

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Answer

We have
$\text{I}=\int\frac{5}{(\text{x}^2+1)(\text{x}+2)}\text{ dx}$
Let $\frac{5}{(\text{x}^2+1)(\text{x}+2)}=\frac{\text{A}}{\text{x}+2}+\frac{\text{Bx}+\text{C}}{\text{x}^2+1}$
$\Rightarrow\frac{5}{(\text{x}^2+1)(\text{x}+2)}=\frac{\text{A}(\text{x}^2+1)+(\text{Bx}+\text{C})(\text{x}+2)}{(\text{x}+2)(\text{x}^2+1)}$
$\Rightarrow5=\text{A}(\text{x}^2+1)+\text{Bx}^2+2\text{Bx}+\text{Cx}+2\text{C}$
$\Rightarrow5=(\text{A}+\text{B})\text{x}^2+(2\text{B}+\text{C})\text{x}+(\text{A}+2\text{C})$
Equating coefficient of like terms
A + B = 0 ...(1)
2B + C = 0 ...(2)
A + 2C = 5 ...(3)
Solving (1), (2) and (3), we get
A = 1
B = -1
C = 2
$\therefore\frac{5}{(\text{x}+2)(\text{x}^2+1)}=\frac{1}{\text{x}+2}+\Big(\frac{-\text{x}+2}{\text{x}^2+1}\Big)$
$\Rightarrow\int\frac{5\text{dx}}{(\text{x}+2)(\text{x}^2+1)}+\int\frac{\text{dx}}{\text{x}+2}-\int\frac{\text{x dx}}{\text{x}^2+1}+2\int\frac{\text{dx}}{\text{x}^2+1}$
Let $\text{x}^2+1=\text{t}$
$\Rightarrow2\text{xdx}=\text{dt}$
$\Rightarrow\text{x dx}=\frac{\text{dt}}{2}$
$\therefore\text{I}=\int\frac{\text{dx}}{\text{x}+2}-\frac{1}{2}\int\frac{\text{dt}}{\text{t}}+2\int\frac{\text{dx}}{\text{x}^2+1^2}$
$=\log|\text{x}+2|-\frac12\log|\text{t}|+2\tan^{-1}\text{x}+\text{C}$
$=\log|\text{x}+2|-\frac{1}{2}\log|\text{x}^2+2|+2\tan^{-1}\text{x}+\text{C}$

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