# Step-by-step Solution

## Solve the trigonometric integral $\int y^3\sin\left(4y\right)dy$

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### Videos

$-\frac{1}{4}y^3\cos\left(4y\right)+\frac{3}{16}y^{2}\sin\left(4y\right)-\frac{3}{128}\left(-4y\cos\left(4y\right)+\sin\left(4y\right)\right)+C_0$

## Step-by-step explanation

Problem to solve:

$\int\left(y^{3\:}Sin\:4y\right)dy$
1

Solve the integral $\int y^3\sin\left(4y\right)dy$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=4y \\ du=4dy\end{matrix}$
2

Isolate $dy$ in the previous equation

$\frac{du}{4}=dy$

$-\frac{1}{4}y^3\cos\left(4y\right)+\frac{3}{16}y^{2}\sin\left(4y\right)-\frac{3}{128}\left(-4y\cos\left(4y\right)+\sin\left(4y\right)\right)+C_0$
$\int\left(y^{3\:}Sin\:4y\right)dy$

### Main topic:

Trigonometric integrals

11. See formulas

~ 1.02 seconds