What is the integral of θ^3sin(θ^2) ?

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

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integral of θ^3sin(θ^2)

\int θ^{3} sin (θ^{2}) d θ

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

Solution steps

\int θ^{3} sin (θ^{2}) d θ

Apply u-substitution

= \int \frac{u sin ( u )}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int u sin (u) d u

Apply Integration By Parts

= \frac{1}{2} (- u cos (u) - \int - cos (u) d u)

\int - cos (u) d u = - sin (u)

= \frac{1}{2} (- u cos (u) - (- sin (u)))

Substitute back

u = θ^{2}

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

Simplify

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

Add a constant to the solution

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

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