What is the integral of sin^4(θ) ?

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

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integral of sin^4(θ)

\int sin^{4} (θ) d θ

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

Solution steps

\int sin^{4} (θ) d θ

Apply Integral Reduction

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{3}{4} sec (x) tan (x) d x

tan (2 x + 3)

x \to 0 lim (3 - 7 x - 5 x^{2})

2 x + 3 y + 4 = 0, [- 4, 4]

\int \frac{sec ^{2} ( x )}{tan ( x )} d x

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