Evaluate the integral: e x ( cos x - sin x ) ? d x

Question

Evaluate the integral: $\int e^{x} (\cos x - \sin x) d x$

Smart Achievers · Accepted Answer

(3)

Ans. $e^{x} \cos x + C$

Explanation :

The given integral is,

$I = \int e^{x} (\cos x - \sin x) d x$

Let, $f (x) = \cos x$

Differentiating $f (x)$ ,

$f' (x) = - \sin x$

Substitute $f (x)$ for $\cos x$ and $f' (x)$ for $- \sin x$ in the given integral.

$I = \int e^{x} (\cos x - \sin x) d x$

$= \int e^{x} {f (x) + f' (x)} d x$

We know that

$\int e^{x} {f (x) + f' (x)} d x = e^{x} f (x) + C$

$∴$ $I = e^{x} \cos x + C$

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