What is the integral of (e^ysin(2x))/(cos(x)) ?

The integral of (e^ysin(2x))/(cos(x)) is -2e^ycos(x)+C

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integral of (e^ysin(2x))/(cos(x))

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

- 2 e^{y} cos (x) + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

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

Take the constant out:

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

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

Use the common integral:

\int sin (x) d x = - cos (x)

= e^{y} \cdot 2 (- cos (x))

Simplify

= - 2 e^{y} cos (x)

Add a constant to the solution

= - 2 e^{y} cos (x) + C

What is the integral of (e^ysin(2x))/(cos(x)) ?
The integral of (e^ysin(2x))/(cos(x)) is -2e^ycos(x)+C