^ ( x + )dx is equal to: - Vidyadip

MCQ

$\int\text{x}^{\sin\text{x}}\Big(\frac{\sin\text{x}}{\text{x}}+\cos\text{x}\cdot\log\text{x}\Big)\text{dx}$ is equal to:

✓
$\text{x}^{\sin\text{x}}+\text{C}$
B
$\text{x}^{\sin\text{x}}\cos\text{x}+\text{C}$
C
$\frac{(\text{x}^{\sin\text{x}})^2}{2}+\text{C}$
D
None of these.

✓

Answer

Correct option: A.

$\text{x}^{\sin\text{x}}+\text{C}$

$\int\text{x}^{\sin\text{x}}\Big(\frac{\sin\text{x}}{\text{x}}+\cos\text{x}\cdot\log\text{x}\Big)\text{dx}$
Put $\text{x}^{\sin\text{x}}=\text{t}$
Taking $\log$ on both sides,
$\log\text{t}=\sin\text{x}\log\text{x}$
$\frac{1}{\text{t}}=\frac{\sin\text{x}}{\text{x}}+\cos\text{x}\log\text{x}$
$1=\int\text{t}\cdot\frac{\text{dt}}{\text{t}}$
$1=\text{t}+\text{C}$
$1=\text{x}^{\sin\text{x}}+\text{C}$

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