What is the integral of 6x^5cos(x^6) ?

The integral of 6x^5cos(x^6) is sin(x^6)+C

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integral of 6x^5cos(x^6)

\int 6 x^{5} cos (x^{6}) d x

sin (x^{6}) + C

Solution steps

\int 6 x^{5} cos (x^{6}) d x

Take the constant out:

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

= 6 \cdot \int x^{5} cos (x^{6}) d x

Apply u-substitution

: \int \frac{cos ( u )}{6} d u

= 6 \cdot \int \frac{cos ( u )}{6} d u

Take the constant out:

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

= 6 \cdot \frac{1}{6} \cdot \int cos (u) d u

Use the common integral:

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

= 6 \cdot \frac{1}{6} sin (u)

Substitute back

u = x^{6}

= 6 \cdot \frac{1}{6} sin (x^{6})

Simplify

6 \cdot \frac{1}{6} sin (x^{6}) : sin (x^{6})

= sin (x^{6})

Add a constant to the solution

= sin (x^{6}) + C

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What is the integral of 6x^5cos(x^6) ?
The integral of 6x^5cos(x^6) is sin(x^6)+C