What is the integral of (θ^3)/(sqrt(9-θ^2)) ?

The integral of (θ^3)/(sqrt(9-θ^2)) is 1/3 (9-θ^2)^{3/2}-9sqrt(9-θ^2)+C

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integral of (θ^3)/(sqrt(9-θ^2))

\int \frac{θ ^{3}}{9 - θ ^{2}} d θ

\frac{1}{3} (9 - θ^{2})^{\frac{3}{2}} - 9 9 - θ^{2} + C

Solution steps

\int \frac{θ ^{3}}{9 - θ ^{2}} d θ

Apply u-substitution

: \int u^{2} - 9 d u

= \int u^{2} - 9 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 u^{2} d u - \int 9 d u

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

\int 9 d u = 9 u

= \frac{u ^{3}}{3} - 9 u

Substitute back

u = 9 - θ^{2}

= \frac{( 9 - θ ^{2} ) ^{3}}{3} - 9 9 - θ^{2}

Simplify

\frac{( 9 - θ ^{2} ) ^{3}}{3} - 9 9 - θ^{2} : \frac{1}{3} (9 - θ^{2})^{\frac{3}{2}} - 9 9 - θ^{2}

= \frac{1}{3} (9 - θ^{2})^{\frac{3}{2}} - 9 9 - θ^{2}

Add a constant to the solution

= \frac{1}{3} (9 - θ^{2})^{\frac{3}{2}} - 9 9 - θ^{2} + C

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What is the integral of (θ^3)/(sqrt(9-θ^2)) ?
The integral of (θ^3)/(sqrt(9-θ^2)) is 1/3 (9-θ^2)^{3/2}-9sqrt(9-θ^2)+C