What is the integral of cos^3(θ) ?

The integral of cos^3(θ) is sin(θ)-(sin^3(θ))/3+C

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integral of cos^3(θ)

\int cos^{3} (θ) d θ

sin (θ) - \frac{sin ^{3} ( θ )}{3} + C

Solution steps

\int cos^{3} (θ) d θ

Rewrite using trig identities

= \int (1 - sin^{2} (θ)) cos (θ) d θ

Apply u-substitution

: \int 1 - u^{2} d u

= \int 1 - u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u - \int u^{2} d u

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= u - \frac{u ^{3}}{3}

Substitute back

u = sin (θ)

= sin (θ) - \frac{sin ^{3} ( θ )}{3}

Add a constant to the solution

= sin (θ) - \frac{sin ^{3} ( θ )}{3} + C

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\int_{6}^{8} (\frac{x ^{2} + 6}{x ^{2}}) d x

\int \frac{x ^{3} - x ^{2} + x - 1}{x ^{2} + 9} d x

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