Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Apply the distributive property.

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

Step 1.5

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

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

Step 1.6

Multiply $2$ by $2$ .

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

Step 1.7

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

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

Step 1.8

Add $2$ and $2$ .

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

Step 1.9

Add $2 y^{2}$ and $2 y^{2}$ .

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

Step 1.10

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

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

Step 1.11

Move $4$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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

Step 7.3

Reorder terms.

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