≤ft(e^{ ( x)}+ x) x d x=

Correct Answer is : x(sin⁡x−cos⁡x)+(sin⁡x+cos⁡x)+cx( x- x)+( x+ x)+cx(sinx−cosx)+(sinx+cosx)+c

Correct Answer is : x(sin⁡x−cos⁡x)+(sin⁡x+cos⁡x)+cx( x- x)+( x+ x)+cx(sinx−cosx)+(sinx+cosx)+c

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MHTCET 20 Apr 2026 Shift 1 Paper

Question : 129 of 150

Marks: +1, -0

\int\left(e^{\log (\sin x)}+\cos x\right) x d x=

Solution:

Concept:

Simplify

e^{\log(\sin x)}

\sin x

, then integrate

x(\sin x + \cos x)

using integration by parts.

Explanation:

Since

\log

is the natural logarithm,

e^{\log(\sin x)} = \sin x

Therefore, the integral becomes

\int x(\sin x + \cos x)\,dx

Split it as

\int x\sin x\,dx + \int x\cos x\,dx

Using integration by parts:

\int x\sin x\,dx = -x\cos x + \sin x

\int x\cos x\,dx = x\sin x + \cos x

Adding both results:

\int x(\sin x + \cos x)\,dx = -x\cos x + \sin x + x\sin x + \cos x

Rearranging terms gives

x(\sin x - \cos x) + (\sin x + \cos x) + C

Answer:

The correct option is D:

x(\sin x - \cos x) + (\sin x + \cos x) + c

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