Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Apply the distributive property.

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

Step 2.6

Apply the distributive property.

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

Step 2.7

Apply the distributive property.

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

Step 2.8

Reorder $x$ and $2$ .

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Move $x^{2}$ .

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Add $2$ and $1$ .

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Add $3$ and $1$ .

$2 \int x^{4} + 2 x^{2} x + 2 x^{2} x + 2 \cdot 2 x^{2} d x$

Step 2.18

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

$2 \int x^{4} + 2 (x^{2} x^{1}) + 2 x^{2} x + 2 \cdot 2 x^{2} d x$

Step 2.19

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

$2 \int x^{4} + 2 x^{2 + 1} + 2 x^{2} x + 2 \cdot 2 x^{2} d x$

Step 2.20

Add $2$ and $1$ .

$2 \int x^{4} + 2 x^{3} + 2 x^{2} x + 2 \cdot 2 x^{2} d x$

Step 2.21

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

$2 \int x^{4} + 2 x^{3} + 2 (x^{2} x^{1}) + 2 \cdot 2 x^{2} d x$

Step 2.22

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

$2 \int x^{4} + 2 x^{3} + 2 x^{2 + 1} + 2 \cdot 2 x^{2} d x$

Step 2.23

Add $2$ and $1$ .

$2 \int x^{4} + 2 x^{3} + 2 x^{3} + 2 \cdot 2 x^{2} d x$

Step 2.24

Multiply $2$ by $2$ .

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

Step 2.25

Add $2 x^{3}$ and $2 x^{3}$ .

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

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

Step 3

Split the single integral into multiple integrals.

$2 (\int x^{4} d x + \int 4 x^{3} d x + \int 4 x^{2} d x)$

Step 4

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

$2 (\frac{1}{5} x^{5} + C + \int 4 x^{3} d x + \int 4 x^{2} d x)$

Step 5

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

$2 (\frac{1}{5} x^{5} + C + 4 \int x^{3} d x + \int 4 x^{2} d x)$

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Reorder terms.

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

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

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$