What is the integral of x^9cos(x^5) ?

The integral of x^9cos(x^5) is 1/5 (x^5sin(x^5)+cos(x^5))+C

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integral of x^9cos(x^5)

\int x^{9} cos (x^{5}) d x

\frac{1}{5} (x^{5} sin (x^{5}) + cos (x^{5})) + C

Solution steps

\int x^{9} cos (x^{5}) d x

Apply u-substitution

= \int \frac{u cos ( u )}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int u cos (u) d u

Apply Integration By Parts

= \frac{1}{5} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{5} (u sin (u) - (- cos (u)))

Substitute back

u = x^{5}

= \frac{1}{5} (x^{5} sin (x^{5}) - (- cos (x^{5})))

Simplify

= \frac{1}{5} (x^{5} sin (x^{5}) + cos (x^{5}))

Add a constant to the solution

= \frac{1}{5} (x^{5} sin (x^{5}) + cos (x^{5})) + C

What is the integral of x^9cos(x^5) ?
The integral of x^9cos(x^5) is 1/5 (x^5sin(x^5)+cos(x^5))+C