What is the integral of e^xsqrt(1-cos^2(x)) ?

The integral of e^xsqrt(1-cos^2(x)) is -(e^xcos(x))/2+(e^xsin(x))/2+C

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integral of e^xsqrt(1-cos^2(x))

\int e^{x} 1 - cos^{2} (x) d x

- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2} + C

Solution steps

\int e^{x} 1 - cos^{2} (x) d x

Simplify

e^{x} 1 - cos^{2} (x) : e^{x} sin (x)

= \int e^{x} sin (x) d x

Apply Integration By Parts

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

Therefore

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

Isolate

\int e^{x} sin (x) d x

= - \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2}

Add a constant to the solution

= - \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2} + C

What is the integral of e^xsqrt(1-cos^2(x)) ?
The integral of e^xsqrt(1-cos^2(x)) is -(e^xcos(x))/2+(e^xsin(x))/2+C