Calculus Examples

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

Step 1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

Tap for more steps...

Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2

Multiply $2$ by $- 1$ .

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

Step 3

Expand ${(2 x^{3} + 3 x)}^{2} x^{- 2}$ .

Tap for more steps...

Step 3.1

Rewrite ${(2 x^{3} + 3 x)}^{2}$ as $(2 x^{3} + 3 x) (2 x^{3} + 3 x)$ .

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Apply the distributive property.

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Apply the distributive property.

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

Step 3.8

Move $x^{3}$ .

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

Step 3.9

Move $x^{3}$ .

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

Step 3.10

Move $x$ .

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

Step 3.11

Move $x$ .

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

Step 3.12

Multiply $2$ by $2$ .

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

Step 3.13

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

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

Step 3.14

Add $3$ and $3$ .

$\int 4 x^{6} x^{- 2} + 2 \cdot 3 x^{3} x \cdot x^{- 2} + 3 \cdot 2 x \cdot x^{3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.15

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

$\int 4 x^{6 - 2} + 2 \cdot 3 x^{3} x \cdot x^{- 2} + 3 \cdot 2 x \cdot x^{3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.16

Subtract $2$ from $6$ .

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

Step 3.17

Multiply $2$ by $3$ .

$\int 4 x^{4} + 6 x^{3} x \cdot x^{- 2} + 3 \cdot 2 x \cdot x^{3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.18

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

$\int 4 x^{4} + 6 (x^{3} x^{1}) x^{- 2} + 3 \cdot 2 x \cdot x^{3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.19

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

$\int 4 x^{4} + 6 x^{3 + 1} x^{- 2} + 3 \cdot 2 x \cdot x^{3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.20

Add $3$ and $1$ .

$\int 4 x^{4} + 6 x^{4} x^{- 2} + 3 \cdot 2 x \cdot x^{3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.21

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

$\int 4 x^{4} + 6 x^{4 - 2} + 3 \cdot 2 x \cdot x^{3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.22

Subtract $2$ from $4$ .

$\int 4 x^{4} + 6 x^{2} + 3 \cdot 2 x \cdot x^{3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.23

Multiply $3$ by $2$ .

$\int 4 x^{4} + 6 x^{2} + 6 x \cdot x^{3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.24

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

$\int 4 x^{4} + 6 x^{2} + 6 (x^{1} x^{3}) x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.25

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

$\int 4 x^{4} + 6 x^{2} + 6 x^{1 + 3} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.26

Add $1$ and $3$ .

$\int 4 x^{4} + 6 x^{2} + 6 x^{4} x^{- 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.27

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

$\int 4 x^{4} + 6 x^{2} + 6 x^{4 - 2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.28

Subtract $2$ from $4$ .

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 3 \cdot 3 x \cdot x \cdot x^{- 2} d x$

Step 3.29

Multiply $3$ by $3$ .

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 9 x \cdot x \cdot x^{- 2} d x$

Step 3.30

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

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 9 (x^{1} x) x^{- 2} d x$

Step 3.31

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

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 9 (x^{1} x^{1}) x^{- 2} d x$

Step 3.32

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

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 9 x^{1 + 1} x^{- 2} d x$

Step 3.33

Add $1$ and $1$ .

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 9 x^{2} x^{- 2} d x$

Step 3.34

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

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 9 x^{2 - 2} d x$

Step 3.35

Subtract $2$ from $2$ .

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 9 x^{0} d x$

Step 3.36

Anything raised to $0$ is $1$ .

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 9 \cdot 1 d x$

Step 3.37

Multiply $9$ by $1$ .

$\int 4 x^{4} + 6 x^{2} + 6 x^{2} + 9 d x$

Step 3.38

Add $6 x^{2}$ and $6 x^{2}$ .

$\int 4 x^{4} + 12 x^{2} + 9 d x$

Step 4

Split the single integral into multiple integrals.

$\int 4 x^{4} d x + \int 12 x^{2} d x + \int 9 d x$

Step 5

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

$4 \int x^{4} d x + \int 12 x^{2} d x + \int 9 d x$

Step 6

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

$4 (\frac{1}{5} x^{5} + C) + \int 12 x^{2} d x + \int 9 d x$

Step 7

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$4 (\frac{1}{5} x^{5} + C) + 12 \int x^{2} d x + \int 9 d x$

Step 8

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

$4 (\frac{1}{5} x^{5} + C) + 12 (\frac{1}{3} x^{3} + C) + \int 9 d x$

Step 9

Apply the constant rule.

$4 (\frac{1}{5} x^{5} + C) + 12 (\frac{1}{3} x^{3} + C) + 9 x + C$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Simplify.

Tap for more steps...

Step 10.1.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$4 (\frac{x^{5}}{5} + C) + 12 (\frac{1}{3} x^{3} + C) + 9 x + C$

Step 10.1.2

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

$4 (\frac{x^{5}}{5} + C) + 12 (\frac{x^{3}}{3} + C) + 9 x + C$

Step 10.2

Simplify.

$\frac{4 x^{5}}{5} + 4 x^{3} + 9 x + C$

Step 10.3

Reorder terms.

$\frac{4}{5} x^{5} + 4 x^{3} + 9 x + C$