What is the integral of (e^xsin(x))/(sec(x)) ?

The integral of (e^xsin(x))/(sec(x)) is -(e^xcos(2x))/5+(e^xsin(2x))/(10)+C

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integral of (e^xsin(x))/(sec(x))

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

- \frac{e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{10} + C

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{sec ( x )} = cos (x)

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

Rewrite using trig identities

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

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{x} sin (2 x) d x

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

: e^{x} \frac{1}{2} sin (2 x) - \int e^{x} \frac{1}{2} sin (2 x) d x

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

Therefore

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

Isolate

\int e^{x} \frac{1}{2} sin (2 x) d x

= - \frac{e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{10}

Add a constant to the solution

= - \frac{e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{10} + C

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What is the integral of (e^xsin(x))/(sec(x)) ?
The integral of (e^xsin(x))/(sec(x)) is -(e^xcos(2x))/5+(e^xsin(2x))/(10)+C