^2 dx is equal to: - Vidyadip

MCQ

$\int\text{x}\sec\text{x}^2\text{ dx}$ is equal to:

✓
$\frac{1}{2}\log\big(\sec\text{x}^2+\tan\text{x}^2\big)+\text{C}$
B
$\frac{\text{x}^2}{2}\log\big(\sec\text{x}^2+\tan\text{x}^2\big)+\text{C}$
C
$2\log\big(\sec\text{x}^2+\tan\text{x}^2\big)+\text{C}$
D
none of these.

✓

Answer

Correct option: A.

$\frac{1}{2}\log\big(\sec\text{x}^2+\tan\text{x}^2\big)+\text{C}$

$\text{I}=\int\text{x}\sec\text{x}^2\text{ dx}$
Put $\text{x}^2=\text{t}$
$=\text{x}=\sqrt{\text{t}}$
$2\text{xdx}=\text{dt}$
$\text{xdx}=\frac{\text{dt}}{2}$
$\text{I}=\int\sec\text{t}\frac{\text{dt}}{2}$
$\text{I}=\frac{1}{2}\log(\sec\text{t}+\tan\text{t})+\text{C}$
$\frac{1}{2}\log\big(\sec\text{x}^2+\tan\text{x}^2\big)+\text{C}$

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