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

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

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

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

sin (x^{5}) + C

Solution steps

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

Apply u-substitution

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

= 5 \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

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

Use the common integral:

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

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

Substitute back

u = x^{5}

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

Simplify

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

= sin (x^{5})

Add a constant to the solution

= sin (x^{5}) + C

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