What is the integral of x^7sin(2x^4) ?

The integral of x^7sin(2x^4) is 1/16 (-2x^4cos(2x^4)+sin(2x^4))+C

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integral of x^7sin(2x^4)

\int x^{7} sin (2 x^{4}) d x

\frac{1}{16} (- 2 x^{4} cos (2 x^{4}) + sin (2 x^{4})) + C

Solution steps

\int x^{7} sin (2 x^{4}) d x

Apply u-substitution

= \int \frac{u sin ( u )}{16} d u

Take the constant out:

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

= \frac{1}{16} \cdot \int u sin (u) d u

Apply Integration By Parts

= \frac{1}{16} (- u cos (u) - \int - cos (u) d u)

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

= \frac{1}{16} (- u cos (u) - (- sin (u)))

Substitute back

u = 2 x^{4}

= \frac{1}{16} (- 2 x^{4} cos (2 x^{4}) - (- sin (2 x^{4})))

Simplify

= \frac{1}{16} (- 2 x^{4} cos (2 x^{4}) + sin (2 x^{4}))

Add a constant to the solution

= \frac{1}{16} (- 2 x^{4} cos (2 x^{4}) + sin (2 x^{4})) + C

What is the integral of x^7sin(2x^4) ?
The integral of x^7sin(2x^4) is 1/16 (-2x^4cos(2x^4)+sin(2x^4))+C