What is the integral of x^5sin(x^3) ?

The integral of x^5sin(x^3) is 1/3 (-x^3cos(x^3)+sin(x^3))+C

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integral of x^5sin(x^3)

\int x^{5} sin (x^{3}) d x

\frac{1}{3} (- x^{3} cos (x^{3}) + sin (x^{3})) + C

Solution steps

\int x^{5} sin (x^{3}) d x

Apply u-substitution

= \int \frac{u sin ( u )}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u sin (u) d u

Apply Integration By Parts

= \frac{1}{3} (- u cos (u) - \int - cos (u) d u)

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

= \frac{1}{3} (- u cos (u) - (- sin (u)))

Substitute back

u = x^{3}

= \frac{1}{3} (- x^{3} cos (x^{3}) - (- sin (x^{3})))

Simplify

= \frac{1}{3} (- x^{3} cos (x^{3}) + sin (x^{3}))

Add a constant to the solution

= \frac{1}{3} (- x^{3} cos (x^{3}) + sin (x^{3})) + C

What is the integral of x^5sin(x^3) ?
The integral of x^5sin(x^3) is 1/3 (-x^3cos(x^3)+sin(x^3))+C