Evaluate the Integral integral of (3y^3-2y^2+(e^y)/6) with respect to y

$\int (3 y^{3} - 2 y^{2} + \frac{e^{y}}{6}) d y$

Step 1

Remove parentheses.

$\int 3 y^{3} - 2 y^{2} + \frac{e^{y}}{6} d y$

Step 2

Split the single integral into multiple integrals.

$\int 3 y^{3} d y + \int - 2 y^{2} d y + \int \frac{e^{y}}{6} d y$

Step 3

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$3 \int y^{3} d y + \int - 2 y^{2} d y + \int \frac{e^{y}}{6} d y$

Step 4

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

$3 (\frac{1}{4} y^{4} + C) + \int - 2 y^{2} d y + \int \frac{e^{y}}{6} d y$

Step 5

Combine $\frac{1}{4}$ and $y^{4}$ .

$3 (\frac{y^{4}}{4} + C) + \int - 2 y^{2} d y + \int \frac{e^{y}}{6} d y$

Step 6

Since $- 2$ is constant with respect to $y$ , move $- 2$ out of the integral.

$3 (\frac{y^{4}}{4} + C) - 2 \int y^{2} d y + \int \frac{e^{y}}{6} d y$

Step 7

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$3 (\frac{y^{4}}{4} + C) - 2 (\frac{1}{3} y^{3} + C) + \int \frac{e^{y}}{6} d y$

Step 8

Combine $\frac{1}{3}$ and $y^{3}$ .

$3 (\frac{y^{4}}{4} + C) - 2 (\frac{y^{3}}{3} + C) + \int \frac{e^{y}}{6} d y$

Step 9

Since $\frac{1}{6}$ is constant with respect to $y$ , move $\frac{1}{6}$ out of the integral.

$3 (\frac{y^{4}}{4} + C) - 2 (\frac{y^{3}}{3} + C) + \frac{1}{6} \int e^{y} d y$

Step 10

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

$3 (\frac{y^{4}}{4} + C) - 2 (\frac{y^{3}}{3} + C) + \frac{1}{6} (e^{y} + C)$

Step 11

Simplify.

$\frac{3 y^{4}}{4} - \frac{2 y^{3}}{3} + \frac{1}{6} e^{y} + C$

Step 12

Reorder terms.

$\frac{3}{4} y^{4} - \frac{2}{3} y^{3} + \frac{1}{6} e^{y} + C$