What is the integral of t^3sin(t^4) ?

The integral of t^3sin(t^4) is -1/4 cos(t^4)+C

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integral of t^3sin(t^4)

\int t^{3} sin (t^{4}) d t

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

Solution steps

\int t^{3} sin (t^{4}) d t

Apply u-substitution

: \int \frac{sin ( u )}{4} d u

= \int \frac{sin ( u )}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{4} (- cos (u))

Substitute back

u = t^{4}

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

Simplify

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

Add a constant to the solution

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

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What is the integral of t^3sin(t^4) ?
The integral of t^3sin(t^4) is -1/4 cos(t^4)+C