x^3 x+1 dx is equal to: - Vidyadip

MCQ

$\int\frac{\text{x}^3}{\text{x}+1}\text{ dx}$ is equal to:

A
$\text{x}+\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
B
$\text{x}+\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
C
$\text{x}-\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
✓
$\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$

✓

Answer

Correct option: D.

$\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$

$\text{I}=\int\frac{\text{x}^3}{\text{x}+1}\text{ dx}$$\text{I}=\int\frac{\text{x}^3+1-1}{\text{x}+1}\text{ dx}$
$\text{I}=\int\frac{(\text{x}+1)(\text{x}^2-\text{x}+1)}{\text{x}+1}\text{ dx}$
$\text{I}=\int\Big(\text{x}^2-\text{x}+1-\frac{1}{\text{x}+1}\Big)\text{dx}$
$\text{I}=\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$

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