What is the integral of sin^3(5x)cos(5x) ?

The integral of sin^3(5x)cos(5x) is 1/20 sin^4(5x)+C

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integral of sin^3(5x)cos(5x)

\int sin^{3} (5 x) cos (5 x) d x

\frac{1}{20} sin^{4} (5 x) + C

Solution steps

\int sin^{3} (5 x) cos (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{3}{4} u^{\frac{4}{3}}

= \frac{1}{15} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = sin^{3} (5 x)

= \frac{1}{15} \cdot \frac{3}{4} (sin^{3} (5 x))^{\frac{4}{3}}

Simplify

\frac{1}{15} \cdot \frac{3}{4} (sin^{3} (5 x))^{\frac{4}{3}} : \frac{1}{20} sin^{4} (5 x)

= \frac{1}{20} sin^{4} (5 x)

Add a constant to the solution

= \frac{1}{20} sin^{4} (5 x) + C

\int \frac{1 + x ^{2}}{x ^{3}} d x

\int \frac{x ^{2}}{1 - x ^{4}} d x

\int \frac{x}{x ^{2} + 2 x + 3} d x

\int \frac{x ^{2} + 4 x + 27}{( x - 2 ) ( x ^{2} + 9 )} d x

\frac{1}{x ^{3}}

What is the integral of sin^3(5x)cos(5x) ?
The integral of sin^3(5x)cos(5x) is 1/20 sin^4(5x)+C