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

The integral of cos((x^5)/5)sin((x^5)/5)x^4 is -1/2 cos^2((x^5)/5)+C

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

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

- \frac{1}{2} cos^{2} (\frac{x ^{5}}{5}) + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int - 5 v d v

= \frac{1}{5} \cdot \int - 5 v d v

Take the constant out:

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

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

Apply the Power Rule

: \frac{v ^{2}}{2}

= \frac{1}{5} (- 5 \cdot \frac{v ^{2}}{2})

Substitute back

= \frac{1}{5} - 5 \cdot \frac{cos ^{2} ( \frac{x ^{5}}{5} )}{2}

Simplify

\frac{1}{5} - 5 \cdot \frac{cos ^{2} ( \frac{x ^{5}}{5} )}{2} : - \frac{1}{2} cos^{2} (\frac{x ^{5}}{5})

= - \frac{1}{2} cos^{2} (\frac{x ^{5}}{5})

Add a constant to the solution

= - \frac{1}{2} cos^{2} (\frac{x ^{5}}{5}) + C

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What is the integral of cos((x^5)/5)sin((x^5)/5)x^4 ?
The integral of cos((x^5)/5)sin((x^5)/5)x^4 is -1/2 cos^2((x^5)/5)+C