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

The integral of sin^3(x)sin(x) is -sin^3(x)cos(x)+3/8 (x-1/4 sin(4x))+C

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

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

- sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)) + C

Solution steps

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

Apply Integration By Parts

: - sin^{3} (x) cos (x) - \int - 3 sin^{2} (x) cos^{2} (x) d x

= - sin^{3} (x) cos (x) - \int - 3 sin^{2} (x) cos^{2} (x) d x

\int - 3 sin^{2} (x) cos^{2} (x) d x = - \frac{3}{8} (x - \frac{1}{4} sin (4 x))

= - sin^{3} (x) cos (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))

Simplify

= - sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))

Add a constant to the solution

= - sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)) + C

\int \frac{1}{( x ^{2} - 4 x + 6 ) ^{2}} d x

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

\int \frac{1}{1 - 5 x} d x

\int \frac{35}{1 + 25 x ^{2}} d x

\int \frac{1}{s ^{3}} d s

What is the integral of sin^3(x)sin(x) ?
The integral of sin^3(x)sin(x) is -sin^3(x)cos(x)+3/8 (x-1/4 sin(4x))+C