# Step-by-step Solution

## Solve the trigonometric integral $\int\frac{1}{2}y^2\cos\left(y\right)dy$

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### Videos

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

## Step-by-step explanation

Problem to solve:

$\:\int\left(cos\:y\right)\cdot\frac{1}{2}y^2dy$
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The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$\frac{1}{2}\int y^2\cos\left(y\right)dy$
2

Use the integration by parts theorem to calculate the integral $\int y^2\cos\left(y\right)dy$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$

$\left(-1+\frac{1}{2}y^2\right)\sin\left(y\right)+y\cos\left(y\right)+C_0$
$\:\int\left(cos\:y\right)\cdot\frac{1}{2}y^2dy$