Calculus Examples

$\int x^{2} {(2 x + 5)}^{2} d x$

Step 1

Expand $x^{2} {(2 x + 5)}^{2}$ .

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Step 1.1

Rewrite ${(2 x + 5)}^{2}$ as $(2 x + 5) (2 x + 5)$ .

$\int x^{2} ((2 x + 5) (2 x + 5)) d x$

Step 1.2

Apply the distributive property.

$\int x^{2} (2 x (2 x + 5) + 5 (2 x + 5)) d x$

Step 1.3

Apply the distributive property.

$\int x^{2} (2 x (2 x) + 2 x \cdot 5 + 5 (2 x + 5)) d x$

Step 1.4

Apply the distributive property.

$\int x^{2} (2 x (2 x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) d x$

Step 1.5

Apply the distributive property.

$\int x^{2} (2 x (2 x) + 2 x \cdot 5) + x^{2} (5 (2 x) + 5 \cdot 5) d x$

Step 1.6

Apply the distributive property.

$\int x^{2} (2 x (2 x)) + x^{2} (2 x \cdot 5) + x^{2} (5 (2 x) + 5 \cdot 5) d x$

Step 1.7

Apply the distributive property.

$\int x^{2} (2 x (2 x)) + x^{2} (2 x \cdot 5) + x^{2} (5 (2 x)) + x^{2} (5 \cdot 5) d x$

Step 1.8

Move $x$ .

$\int x^{2} (2 \cdot 2 x \cdot x) + x^{2} (2 x \cdot 5) + x^{2} (5 (2 x)) + x^{2} (5 \cdot 5) d x$

Step 1.9

Move parentheses.

$\int x^{2} (2 \cdot 2 x) x + x^{2} (2 x \cdot 5) + x^{2} (5 (2 x)) + x^{2} (5 \cdot 5) d x$

Step 1.10

Move parentheses.

$\int x^{2} (2 \cdot 2) x \cdot x + x^{2} (2 x \cdot 5) + x^{2} (5 (2 x)) + x^{2} (5 \cdot 5) d x$

Step 1.11

Move $x^{2}$ .

$\int 2 \cdot 2 x^{2} x \cdot x + x^{2} (2 x \cdot 5) + x^{2} (5 (2 x)) + x^{2} (5 \cdot 5) d x$

Step 1.12

Move $x$ .

$\int 2 \cdot 2 x^{2} x \cdot x + x^{2} (2 \cdot 5 x) + x^{2} (5 (2 x)) + x^{2} (5 \cdot 5) d x$

Step 1.13

Move parentheses.

$\int 2 \cdot 2 x^{2} x \cdot x + x^{2} (2 \cdot 5) x + x^{2} (5 (2 x)) + x^{2} (5 \cdot 5) d x$

Step 1.14

Move $x^{2}$ .

$\int 2 \cdot 2 x^{2} x \cdot x + 2 \cdot 5 x^{2} x + x^{2} (5 (2 x)) + x^{2} (5 \cdot 5) d x$

Step 1.15

Move parentheses.

$\int 2 \cdot 2 x^{2} x \cdot x + 2 \cdot 5 x^{2} x + x^{2} (5 \cdot 2) x + x^{2} (5 \cdot 5) d x$

Step 1.16

Move $x^{2}$ .

$\int 2 \cdot 2 x^{2} x \cdot x + 2 \cdot 5 x^{2} x + 5 \cdot 2 x^{2} x + x^{2} (5 \cdot 5) d x$

Step 1.17

Move $x^{2}$ .

$\int 2 \cdot 2 x^{2} x \cdot x + 2 \cdot 5 x^{2} x + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.18

Multiply $2$ by $2$ .

$\int 4 x^{2} x \cdot x + 2 \cdot 5 x^{2} x + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.19

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

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

Step 1.20

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

$\int 4 x^{2 + 1} x + 2 \cdot 5 x^{2} x + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.21

Add $2$ and $1$ .

$\int 4 x^{3} x + 2 \cdot 5 x^{2} x + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.22

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

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

Step 1.23

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

$\int 4 x^{3 + 1} + 2 \cdot 5 x^{2} x + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.24

Add $3$ and $1$ .

$\int 4 x^{4} + 2 \cdot 5 x^{2} x + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.25

Multiply $2$ by $5$ .

$\int 4 x^{4} + 10 x^{2} x + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.26

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

$\int 4 x^{4} + 10 (x^{2} x^{1}) + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.27

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

$\int 4 x^{4} + 10 x^{2 + 1} + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.28

Add $2$ and $1$ .

$\int 4 x^{4} + 10 x^{3} + 5 \cdot 2 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.29

Multiply $5$ by $2$ .

$\int 4 x^{4} + 10 x^{3} + 10 x^{2} x + 5 \cdot 5 x^{2} d x$

Step 1.30

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

$\int 4 x^{4} + 10 x^{3} + 10 (x^{2} x^{1}) + 5 \cdot 5 x^{2} d x$

Step 1.31

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

$\int 4 x^{4} + 10 x^{3} + 10 x^{2 + 1} + 5 \cdot 5 x^{2} d x$

Step 1.32

Add $2$ and $1$ .

$\int 4 x^{4} + 10 x^{3} + 10 x^{3} + 5 \cdot 5 x^{2} d x$

Step 1.33

Multiply $5$ by $5$ .

$\int 4 x^{4} + 10 x^{3} + 10 x^{3} + 25 x^{2} d x$

Step 1.34

Add $10 x^{3}$ and $10 x^{3}$ .

$\int 4 x^{4} + 20 x^{3} + 25 x^{2} d x$

Step 2

Split the single integral into multiple integrals.

$\int 4 x^{4} d x + \int 20 x^{3} d x + \int 25 x^{2} d x$

Step 3

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

$4 \int x^{4} d x + \int 20 x^{3} d x + \int 25 x^{2} d x$

Step 4

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

$4 (\frac{1}{5} x^{5} + C) + \int 20 x^{3} d x + \int 25 x^{2} d x$

Step 5

Since $20$ is constant with respect to $x$ , move $20$ out of the integral.

$4 (\frac{1}{5} x^{5} + C) + 20 \int x^{3} d x + \int 25 x^{2} d x$

Step 6

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

$4 (\frac{1}{5} x^{5} + C) + 20 (\frac{1}{4} x^{4} + C) + \int 25 x^{2} d x$

Step 7

Since $25$ is constant with respect to $x$ , move $25$ out of the integral.

$4 (\frac{1}{5} x^{5} + C) + 20 (\frac{1}{4} x^{4} + C) + 25 \int x^{2} d x$

Step 8

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

$4 (\frac{1}{5} x^{5} + C) + 20 (\frac{1}{4} x^{4} + C) + 25 (\frac{1}{3} x^{3} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{4 x^{5}}{5} + 5 x^{4} + 25 (\frac{1}{3} x^{3}) + C$

Step 9.2

Simplify.

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Step 9.2.1

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

$\frac{4 x^{5}}{5} + 5 x^{4} + 25 \frac{x^{3}}{3} + C$

Step 9.2.2

Combine $25$ and $\frac{x^{3}}{3}$ .

$\frac{4 x^{5}}{5} + 5 x^{4} + \frac{25 x^{3}}{3} + C$

Step 9.3

Reorder terms.

$\frac{4}{5} x^{5} + 5 x^{4} + \frac{25}{3} x^{3} + C$