What is the integral of 5(cos(x))(sin^4(x)) ?

The integral of 5(cos(x))(sin^4(x)) is sin^5(x)+C

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integral of 5(cos(x))(sin^4(x))

\int 5 (cos (x)) (sin^{4} (x)) d x

sin^{5} (x) + C

Solution steps

\int 5 cos (x) sin^{4} (x) d x

Take the constant out:

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

= 5 \cdot \int cos (x) sin^{4} (x) d x

Apply u-substitution

: \int u^{4} d u

= 5 \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= 5 \cdot \frac{u ^{5}}{5}

Substitute back

u = sin (x)

= 5 \cdot \frac{sin ^{5} ( x )}{5}

Simplify

5 \cdot \frac{sin ^{5} ( x )}{5} : sin^{5} (x)

= sin^{5} (x)

Add a constant to the solution

= sin^{5} (x) + C

\int sin^{4} (x) sin (x) d x

\int 2 + 4 - x d x

\int 5 sinh (ln (2) - \frac{x}{3}) d x

\int \frac{sin ( 2 x )}{e ^{- 4 x}} d x

\int coth (1 - 2 x) d x

What is the integral of 5(cos(x))(sin^4(x)) ?
The integral of 5(cos(x))(sin^4(x)) is sin^5(x)+C