What is the integral of sin^4(z) ?

The integral of sin^4(z) is -1/4 sin^3(z)cos(z)+3/8 (z-1/2 sin(2z))+C

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integral of sin^4(z)

\int sin^{4} (z)

- \frac{1}{4} sin^{3} (z) cos (z) + \frac{3}{8} (z - \frac{1}{2} sin (2 z)) + C

Solution steps

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

Apply Integral Reduction

= - \frac{cos ( z ) sin ^{3} ( z )}{4} + \frac{3}{4} \cdot \int sin^{2} (z) d z

\int sin^{2} (z) d z = \frac{1}{2} (z - \frac{1}{2} sin (2 z))

= - \frac{cos ( z ) sin ^{3} ( z )}{4} + \frac{3}{4} \cdot \frac{1}{2} (z - \frac{1}{2} sin (2 z))

Simplify

- \frac{cos ( z ) sin ^{3} ( z )}{4} + \frac{3}{4} \cdot \frac{1}{2} (z - \frac{1}{2} sin (2 z)) : - \frac{1}{4} sin^{3} (z) cos (z) + \frac{3}{8} (z - \frac{1}{2} sin (2 z))

= - \frac{1}{4} sin^{3} (z) cos (z) + \frac{3}{8} (z - \frac{1}{2} sin (2 z))

Add a constant to the solution

= - \frac{1}{4} sin^{3} (z) cos (z) + \frac{3}{8} (z - \frac{1}{2} sin (2 z)) + C

What is the integral of sin^4(z) ?
The integral of sin^4(z) is -1/4 sin^3(z)cos(z)+3/8 (z-1/2 sin(2z))+C