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

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

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

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

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

Solution steps

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

Apply u-substitution

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

= \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

: - u cos (u) - \int - cos (u) d u

= \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 t^{4}

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

Simplify

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

Add a constant to the solution

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

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