What is the integral of 9x^4cos(x^5+6) ?

The integral of 9x^4cos(x^5+6) is 9/5 sin(x^5+6)+C

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integral of 9x^4cos(x^5+6)

\int 9 x^{4} cos (x^{5} + 6) d x

\frac{9}{5} sin (x^{5} + 6) + C

Solution steps

\int 9 x^{4} cos (x^{5} + 6) d x

Take the constant out:

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

= 9 \cdot \int x^{4} cos (x^{5} + 6) d x

Apply u-substitution

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

= 9 \cdot \int \frac{cos ( u )}{5} d u

Take the constant out:

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

= 9 \cdot \frac{1}{5} \cdot \int cos (u) d u

Use the common integral:

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

= 9 \cdot \frac{1}{5} sin (u)

Substitute back

u = x^{5} + 6

= 9 \cdot \frac{1}{5} sin (x^{5} + 6)

Simplify

9 \cdot \frac{1}{5} sin (x^{5} + 6) : \frac{9}{5} sin (x^{5} + 6)

= \frac{9}{5} sin (x^{5} + 6)

Add a constant to the solution

= \frac{9}{5} sin (x^{5} + 6) + C

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What is the integral of 9x^4cos(x^5+6) ?
The integral of 9x^4cos(x^5+6) is 9/5 sin(x^5+6)+C