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

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

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

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

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

Solution steps

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

Apply u-substitution

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

= \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

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

Apply the Power Rule

: - \frac{1}{u}

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

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

\frac{d y}{d x} = 9 x + y

x \to 0 lim ((x - 1)^{2})

\int_{0}^{\infty} x^{2} e^{- x} d x

(3.5) (2.6)

\frac{d}{d x} (g (x) (x)^{2} (x) + x)

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