What is the integral of ((5cos(5x))/(e^{sin(5x))}) ?

The integral of ((5cos(5x))/(e^{sin(5x))}) is -1/(e^{sin(5x))}+C

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integral of ((5cos(5x))/(e^{sin(5x))})

\int (\frac{5 cos ( 5 x )}{e ^{s i n (5 x)}}) d x

- \frac{1}{e ^{s i n (5 x)}} + C

Solution steps

\int \frac{5 cos ( 5 x )}{e ^{s i n (5 x)}} d x

Take the constant out:

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

= 5 \cdot \int \frac{cos ( 5 x )}{e ^{s i n (5 x)}} d x

Apply u-substitution

: \int \frac{1}{5 u ^{2}} d u

= 5 \cdot \int \frac{1}{5 u ^{2}} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= 5 \cdot \frac{1}{5} (- \frac{1}{u})

Substitute back

u = e^{s i n (5 x)}

= 5 \cdot \frac{1}{5} (- \frac{1}{e ^{s i n (5 x)}})

Simplify

5 \cdot \frac{1}{5} (- \frac{1}{e ^{s i n (5 x)}}) : - \frac{1}{e ^{s i n (5 x)}}

= - \frac{1}{e ^{s i n (5 x)}}

Add a constant to the solution

= - \frac{1}{e ^{s i n (5 x)}} + C

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What is the integral of ((5cos(5x))/(e^{sin(5x))}) ?
The integral of ((5cos(5x))/(e^{sin(5x))}) is -1/(e^{sin(5x))}+C