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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.25 Question 46 Maths Textbook Solution.

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Answer: x\: \sin \: x+\cos x-\frac{1}{4}\cos 2x+c

Given: \int \left ( e^{logx}+\sin \: x \right )\cos \: xdx

Hint: Put e^{logx}=x

Solution:\int \left ( e^{logx}+\sin \: x \right )\cos \: xdx

As e^{logx}=x

               \begin{aligned} &I=\int(x+\sin x) \cos x d x \\ &=\int x \cos x d x+\int \sin x \cos x d x \\ &=x \sin x-\int 1 \sin x d x+\frac{1}{2} \int \sin (2 x) d x \\ &=x \sin x-(-\cos x)+\frac{1}{2} \frac{1}{2}(-\cos 2 x)+c \\ &=x \sin x+\cos x-\frac{1}{4} \cos 2 x+c \end{aligned}

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