What is the integral of (cos(4x))/(e^{sin(4x))} ?

The integral of (cos(4x))/(e^{sin(4x))} is -1/(4e^{sin(4x))}+C

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integral of (cos(4x))/(e^{sin(4x))}

\int \frac{cos ( 4 x )}{e ^{s i n (4 x)}} d x

- \frac{1}{4 e ^{s i n (4 x)}} + C

Solution steps

\int \frac{cos ( 4 x )}{e ^{s i n (4 x)}} d x

Apply u-substitution

= \int \frac{1}{4 u ^{2}} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

= \frac{1}{4} (- \frac{1}{u})

Substitute back

u = e^{s i n (4 x)}

= \frac{1}{4} (- \frac{1}{e ^{s i n (4 x)}})

Simplify

\frac{1}{4} (- \frac{1}{e ^{s i n (4 x)}}) : - \frac{1}{4 e ^{s i n (4 x)}}

= - \frac{1}{4 e ^{s i n (4 x)}}

Add a constant to the solution

= - \frac{1}{4 e ^{s i n (4 x)}} + C

What is the integral of (cos(4x))/(e^{sin(4x))} ?
The integral of (cos(4x))/(e^{sin(4x))} is -1/(4e^{sin(4x))}+C