What is the integral of x^4sin(x^5+1) ?

The integral of x^4sin(x^5+1) is -1/5 cos(x^5+1)+C

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integral of x^4sin(x^5+1)

\int x^{4} sin (x^{5} + 1) d x

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

Solution steps

\int x^{4} sin (x^{5} + 1) d x

Apply u-substitution

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

Use the common integral:

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

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

Substitute back

u = x^{5} + 1

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

Simplify

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

Add a constant to the solution

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

What is the integral of x^4sin(x^5+1) ?
The integral of x^4sin(x^5+1) is -1/5 cos(x^5+1)+C