What is the integral of 10cos(5x)sin^5(5x) ?

The integral of 10cos(5x)sin^5(5x) is 1/3 sin^6(5x)+C

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integral of 10cos(5x)sin^5(5x)

\int 10 cos (5 x) sin^{5} (5 x) d x

\frac{1}{3} sin^{6} (5 x) + C

Solution steps

\int 10 cos (5 x) sin^{5} (5 x) d x

Take the constant out:

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

= 10 \cdot \int cos (5 x) sin^{5} (5 x) d x

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

: \frac{5}{6} u^{\frac{6}{5}}

= 10 \cdot \frac{1}{25} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = sin^{5} (5 x)

= 10 \cdot \frac{1}{25} \cdot \frac{5}{6} (sin^{5} (5 x))^{\frac{6}{5}}

Simplify

10 \cdot \frac{1}{25} \cdot \frac{5}{6} (sin^{5} (5 x))^{\frac{6}{5}} : \frac{1}{3} sin^{6} (5 x)

= \frac{1}{3} sin^{6} (5 x)

Add a constant to the solution

= \frac{1}{3} sin^{6} (5 x) + C

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What is the integral of 10cos(5x)sin^5(5x) ?
The integral of 10cos(5x)sin^5(5x) is 1/3 sin^6(5x)+C