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

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

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

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

- cos (x^{5}) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 5 \cdot \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

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

Use the common integral:

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

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

Substitute back

u = x^{5}

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

Simplify

5 \cdot \frac{1}{5} (- cos (x^{5})) : - cos (x^{5})

= - cos (x^{5})

Add a constant to the solution

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

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