Evaluate the following integrals:


Evaluate the following integrals:

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


Let $\mathrm{I}=\int\left(\mathrm{e}^{\log \mathrm{x}}+\sin \mathrm{x}\right) \cos \mathrm{x} \mathrm{dx}$

$=\int(x+\sin x) \cos x d x$

$=\int x \cos x d x+\int \sin x \cos x d x$

Using integration by parts,

$=x \int \cos x d x-\int \frac{d}{d x} x \int \cos x d x+\frac{1}{2} \int \sin 2 x d x$

$=x \times \sin x-\int \sin x d x+\frac{1}{2}\left(\frac{-\cos 2 x}{2}\right)+c$

$=x \sin x+\cos x-\frac{1}{4} \cos 2 x+c$

$=x \sin x+\cos x-\frac{1}{4}\left[1-2 \sin ^{2} x\right]+c$

$I=x \sin x+\cos x-\frac{1}{4}+\frac{1}{2} \sin ^{2} x+c$

$I=x \sin x+\cos x+\frac{1}{2} \sin ^{2} x+c-\frac{1}{4}$

$I=x \sin x+\cos x+\frac{1}{2} \sin ^{2} x+k$ where, $k=c-\frac{1}{4}$

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