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

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

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

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

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

Solution steps

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

Apply u-substitution

= \int \frac{u cos ( 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 cos (u) d u

Apply Integration By Parts

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

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

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

Substitute back

u = x^{7}

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

Simplify

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

Add a constant to the solution

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

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