What is the integral of 8x^7sin(x^8) ?

The integral of 8x^7sin(x^8) is -cos(x^8)+C

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integral of 8x^7sin(x^8)

\int 8 x^{7} sin (x^{8}) d x

- cos (x^{8}) + C

Solution steps

\int 8 x^{7} sin (x^{8}) d x

Take the constant out:

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

= 8 \cdot \int x^{7} sin (x^{8}) d x

Apply u-substitution

: \int \frac{sin ( u )}{8} d u

= 8 \cdot \int \frac{sin ( u )}{8} d u

Take the constant out:

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

= 8 \cdot \frac{1}{8} \cdot \int sin (u) d u

Use the common integral:

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

= 8 \cdot \frac{1}{8} (- cos (u))

Substitute back

u = x^{8}

= 8 \cdot \frac{1}{8} (- cos (x^{8}))

Simplify

8 \cdot \frac{1}{8} (- cos (x^{8})) : - cos (x^{8})

= - cos (x^{8})

Add a constant to the solution

= - cos (x^{8}) + C

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What is the integral of 8x^7sin(x^8) ?
The integral of 8x^7sin(x^8) is -cos(x^8)+C