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

Question

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

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Answer

$\int\sqrt{\text{e}^\text{x}-1}\text{ dx}$

Let $\text{I}=\int\frac{1}{(\text{x}+1)(\text{x}^2+2\text{x}+2)}\text{ dx}$

$=\int\frac{1}{(\text{x}+1)\big((\text{x}+1)^2+\text{1}\big)}\text{ dx}$

Let $\text{x}+1=\tan\text{u}$

$\Rightarrow\text{dx}=\sec^2\text{u du}$

$\therefore\ \text{I}=\int\frac{\sec^2\text{u}}{\tan\text{u}(\tan^2\text{u}+1)}\text{ du}$

$=\int\frac{\cos\text{u}}{\sin\text{u}}\text{ du}$

$=\log|\sin\text{u}|+\text{C}$

$=\log\Big|\frac{\tan\text{u}}{\sec^2\text{u}}\Big|+\text{C}$

$=\log\bigg|\frac{\text{x}+1}{\sqrt{\text{x}^2+2\text{x}+2}}\bigg|+\text{C}$

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