Calculus Examples

$\int y^{2} {(1 + y)}^{2} d y$

Step 1

Expand $y^{2} {(1 + y)}^{2}$ .

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Step 1.1

Rewrite ${(1 + y)}^{2}$ as $(1 + y) (1 + y)$ .

$\int y^{2} ((1 + y) (1 + y)) d y$

Step 1.2

Apply the distributive property.

$\int y^{2} (1 (1 + y) + y (1 + y)) d y$

Step 1.3

Apply the distributive property.

$\int y^{2} (1 \cdot 1 + 1 y + y (1 + y)) d y$

Step 1.4

Apply the distributive property.

$\int y^{2} (1 \cdot 1 + 1 y + y \cdot 1 + y \cdot y) d y$

Step 1.5

Apply the distributive property.

$\int y^{2} (1 \cdot 1 + 1 y) + y^{2} (y \cdot 1 + y \cdot y) d y$

Step 1.6

Apply the distributive property.

$\int y^{2} (1 \cdot 1) + y^{2} (1 y) + y^{2} (y \cdot 1 + y \cdot y) d y$

Step 1.7

Apply the distributive property.

$\int y^{2} (1 \cdot 1) + y^{2} (1 y) + y^{2} (y \cdot 1) + y^{2} (y \cdot y) d y$

Step 1.8

Move $y^{2}$ .

$\int 1 \cdot 1 y^{2} + y^{2} (1 y) + y^{2} (y \cdot 1) + y^{2} (y \cdot y) d y$

Step 1.9

Reorder $y^{2}$ and $1$ .

$\int 1 \cdot 1 y^{2} + 1 \cdot y^{2} y + y^{2} (y \cdot 1) + y^{2} (y \cdot y) d y$

Step 1.10

Reorder $y$ and $1$ .

$\int 1 \cdot 1 y^{2} + 1 \cdot y^{2} y + y^{2} (1 \cdot y) + y^{2} (y \cdot y) d y$

Step 1.11

Reorder $y^{2}$ and $1$ .

$\int 1 \cdot 1 y^{2} + 1 \cdot y^{2} y + 1 \cdot y^{2} \cdot y + y^{2} (y \cdot y) d y$

Step 1.12

Multiply $1$ by $1$ .

$\int 1 y^{2} + 1 y^{2} y + 1 y^{2} y + y^{2} y \cdot y d y$

Step 1.13

Multiply $y^{2}$ by $1$ .

$\int y^{2} + 1 y^{2} y + 1 y^{2} y + y^{2} y \cdot y d y$

Step 1.14

Multiply $y^{2}$ by $1$ .

$\int y^{2} + y^{2} y + 1 y^{2} y + y^{2} y \cdot y d y$

Step 1.15

Raise $y$ to the power of $1$ .

$\int y^{2} + y^{2} y^{1} + 1 y^{2} y + y^{2} y \cdot y d y$

Step 1.16

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int y^{2} + y^{2 + 1} + 1 y^{2} y + y^{2} y \cdot y d y$

Step 1.17

Add $2$ and $1$ .

$\int y^{2} + y^{3} + 1 y^{2} y + y^{2} y \cdot y d y$

Step 1.18

Multiply $y^{2}$ by $1$ .

$\int y^{2} + y^{3} + y^{2} y + y^{2} y \cdot y d y$

Step 1.19

Raise $y$ to the power of $1$ .

$\int y^{2} + y^{3} + y^{2} y^{1} + y^{2} y \cdot y d y$

Step 1.20

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int y^{2} + y^{3} + y^{2 + 1} + y^{2} y \cdot y d y$

Step 1.21

Add $2$ and $1$ .

$\int y^{2} + y^{3} + y^{3} + y^{2} y \cdot y d y$

Step 1.22

Raise $y$ to the power of $1$ .

$\int y^{2} + y^{3} + y^{3} + y^{2} y^{1} y d y$

Step 1.23

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int y^{2} + y^{3} + y^{3} + y^{2 + 1} y d y$

Step 1.24

Add $2$ and $1$ .

$\int y^{2} + y^{3} + y^{3} + y^{3} y d y$

Step 1.25

Raise $y$ to the power of $1$ .

$\int y^{2} + y^{3} + y^{3} + y^{3} y^{1} d y$

Step 1.26

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int y^{2} + y^{3} + y^{3} + y^{3 + 1} d y$

Step 1.27

Add $3$ and $1$ .

$\int y^{2} + y^{3} + y^{3} + y^{4} d y$

Step 1.28

Add $y^{3}$ and $y^{3}$ .

$\int y^{2} + 2 y^{3} + y^{4} d y$

Step 1.29

Reorder $2 y^{3}$ and $y^{4}$ .

$\int y^{2} + y^{4} + 2 y^{3} d y$

Step 1.30

Move $y^{2}$ .

$\int y^{4} + 2 y^{3} + y^{2} d y$

Step 2

Split the single integral into multiple integrals.

$\int y^{4} d y + \int 2 y^{3} d y + \int y^{2} d y$

Step 3

By the Power Rule, the integral of $y^{4}$ with respect to $y$ is $\frac{1}{5} y^{5}$ .

$\frac{1}{5} y^{5} + C + \int 2 y^{3} d y + \int y^{2} d y$

Step 4

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

$\frac{1}{5} y^{5} + C + 2 \int y^{3} d y + \int y^{2} d y$

Step 5

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

$\frac{1}{5} y^{5} + C + 2 (\frac{1}{4} y^{4} + C) + \int y^{2} d y$

Step 6

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

$\frac{1}{5} y^{5} + C + 2 (\frac{1}{4} y^{4} + C) + \frac{1}{3} y^{3} + C$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{5} y^{5} + C + 2 (\frac{y^{4}}{4} + C) + \frac{1}{3} y^{3} + C$

Step 7.2

Simplify.

$\frac{y^{5}}{5} + \frac{y^{4}}{2} + \frac{1}{3} y^{3} + C$

Step 7.3

Reorder terms.

$\frac{1}{5} y^{5} + \frac{1}{2} y^{4} + \frac{1}{3} y^{3} + C$