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

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

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

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

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

Solution steps

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

Use the basic trigonometric identity:

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

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

Apply Integration By Parts

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

Apply Integration By Parts

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

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