What is the integral of x^3*sin(x^2) ?

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

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integral of x^3*sin(x^2)

\int x^{3} \cdot sin (x^{2}) d x

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

Solution steps

\int x^{3} sin (x^{2}) d x

Apply u-substitution

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

= \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

: - u cos (u) - \int - cos (u) d u

= \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 = x^{2}

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

Simplify

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

Add a constant to the solution

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

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