What is the integral of θ(θ-2)^3 ?

The integral of θ(θ-2)^3 is (θ^5)/5-(3θ^4)/2+4θ^3-4θ^2+C

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integral of θ(θ-2)^3

\int θ (θ - 2)^{3} d θ

\frac{θ ^{5}}{5} - \frac{3 θ ^{4}}{2} + 4 θ^{3} - 4 θ^{2} + C

Solution steps

\int θ (θ - 2)^{3} d θ

Expand

θ (θ - 2)^{3} : θ^{4} - 6 θ^{3} + 12 θ^{2} - 8 θ

= \int θ^{4} - 6 θ^{3} + 12 θ^{2} - 8 θ d θ

Apply the Sum Rule:

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

= \int θ^{4} d θ - \int 6 θ^{3} d θ + \int 12 θ^{2} d θ - \int 8 θ d θ

\int θ^{4} d θ = \frac{θ ^{5}}{5}

\int 6 θ^{3} d θ = \frac{3 θ ^{4}}{2}

\int 12 θ^{2} d θ = 4 θ^{3}

\int 8 θ d θ = 4 θ^{2}

= \frac{θ ^{5}}{5} - \frac{3 θ ^{4}}{2} + 4 θ^{3} - 4 θ^{2}

Add a constant to the solution

= \frac{θ ^{5}}{5} - \frac{3 θ ^{4}}{2} + 4 θ^{3} - 4 θ^{2} + C

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What is the integral of θ(θ-2)^3 ?
The integral of θ(θ-2)^3 is (θ^5)/5-(3θ^4)/2+4θ^3-4θ^2+C