What is the integral of sin(4x^5+10)*x^4 ?

The integral of sin(4x^5+10)*x^4 is -1/20 cos(4x^5+10)+C

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integral of sin(4x^5+10)*x^4

\int sin (4 x^{5} + 10) \cdot x^{4} d x

- \frac{1}{20} cos (4 x^{5} + 10) + C

Solution steps

\int sin (4 x^{5} + 10) x^{4} d x

Apply u-substitution

= \int \frac{sin ( 4 u + 10 )}{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 (4 u + 10) d u

Apply u-substitution

= \frac{1}{5} \cdot \int sin (v) \frac{1}{4} d v

Take the constant out:

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

= \frac{1}{5} \cdot \frac{1}{4} \cdot \int sin (v) d v

Use the common integral:

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

= \frac{1}{5} \cdot \frac{1}{4} (- cos (v))

Substitute back

= \frac{1}{5} \cdot \frac{1}{4} (- cos (4 x^{5} + 10))

Simplify

\frac{1}{5} \cdot \frac{1}{4} (- cos (4 x^{5} + 10)) : - \frac{1}{20} cos (4 x^{5} + 10)

= - \frac{1}{20} cos (4 x^{5} + 10)

Add a constant to the solution

= - \frac{1}{20} cos (4 x^{5} + 10) + C

What is the integral of sin(4x^5+10)*x^4 ?
The integral of sin(4x^5+10)*x^4 is -1/20 cos(4x^5+10)+C