What is the integral of x^{13}sin(x^7) ?

The integral of x^{13}sin(x^7) is 1/7 (-x^7cos(x^7)+sin(x^7))+C

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integral of x^{13}sin(x^7)

\int x^{13} sin (x^{7}) d x

\frac{1}{7} (- x^{7} cos (x^{7}) + sin (x^{7})) + C

Solution steps

\int x^{13} sin (x^{7}) d x

Apply u-substitution

= \int \frac{u sin ( u )}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int u sin (u) d u

Apply Integration By Parts

= \frac{1}{7} (- u cos (u) - \int - cos (u) d u)

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

= \frac{1}{7} (- u cos (u) - (- sin (u)))

Substitute back

u = x^{7}

= \frac{1}{7} (- x^{7} cos (x^{7}) - (- sin (x^{7})))

Simplify

= \frac{1}{7} (- x^{7} cos (x^{7}) + sin (x^{7}))

Add a constant to the solution

= \frac{1}{7} (- x^{7} cos (x^{7}) + sin (x^{7})) + C

What is the integral of x^{13}sin(x^7) ?
The integral of x^{13}sin(x^7) is 1/7 (-x^7cos(x^7)+sin(x^7))+C