What is the integral of (e^xcos(e^x))/(4+sin^2(e^x)) ?

The integral of (e^xcos(e^x))/(4+sin^2(e^x)) is 1/2 arctan((sin(e^x))/2)+C

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integral of (e^xcos(e^x))/(4+sin^2(e^x))

\int \frac{e ^{x} cos ( e ^{x} )}{4 + sin ^{2} ( e ^{x} )} d x

\frac{1}{2} arctan (\frac{sin ( e ^{x} )}{2}) + C

Solution steps

\int \frac{e ^{x} cos ( e ^{x} )}{4 + sin ^{2} ( e ^{x} )} d x

Apply u-substitution

= \int \frac{cos ( u )}{4 + sin ^{2} ( u )} d u

Apply u-substitution

= \int \frac{1}{4 + v ^{2}} d v

Apply Integral Substitution

= \int \frac{1}{2 ( w ^{2} + 1 )} d w

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{w ^{2} + 1} d w

Use the common integral:

\int \frac{1}{w ^{2} + 1} d w = arctan (w)

= \frac{1}{2} arctan (w)

Substitute back

= \frac{1}{2} arctan (\frac{sin ( e ^{x} )}{2})

Add a constant to the solution

= \frac{1}{2} arctan (\frac{sin ( e ^{x} )}{2}) + C

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What is the integral of (e^xcos(e^x))/(4+sin^2(e^x)) ?
The integral of (e^xcos(e^x))/(4+sin^2(e^x)) is 1/2 arctan((sin(e^x))/2)+C