Evaluate: [ ( x)+ ( x)] d x - Vidyadip

MCQ

Evaluate: $\int[\sin (\log x)+\cos (\log x)] d x$

✓
$x \sin (\log x)+C$
B
$\sin (\log x)+C$
C
$x \cos (\log x)+C$
D
$\cos (\log x)+C$

✓

Answer

Correct option: A.

$x \sin (\log x)+C$

(a) : Let $I=\int[\sin (\log x)+\cos (\log x)] d x$
Put $\log x=t \Rightarrow x=e^t \quad \Rightarrow d x=e^t d t$
$
\begin{aligned}
\therefore \quad I & =\int(\sin t+\cos t) e^t d t=e^t \sin t+C \\
& =x \sin (\log x)+C\left[\because\left[f(x)+f^{\prime}(x)\right] e^x d x=e^x f(x)+C\right)
\end{aligned}
$

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