Calculus Examples

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

Step 1

Expand ${(x^{2} + 1)}^{3} x^{3}$ .

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Rewrite the exponentiation as a product.

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

Step 1.3

Rewrite the exponentiation as a product.

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

Step 1.4

Rewrite the exponentiation as a product.

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

Step 1.5

Rewrite the exponentiation as a product.

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

Step 1.6

Rewrite the exponentiation as a product.

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

Step 1.7

Rewrite the exponentiation as a product.

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

Step 1.8

Apply the distributive property.

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

Step 1.9

Apply the distributive property.

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

Step 1.10

Apply the distributive property.

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

Step 1.11

Move $x^{2}$ .

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

Step 1.12

Move $x^{2}$ .

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

Step 1.13

Move $x^{2}$ .

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

Step 1.14

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

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

Step 1.15

Add $2$ and $2$ .

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

Step 1.16

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

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

Step 1.17

Add $4$ and $2$ .

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

Step 1.18

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

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

Step 1.19

Add $6$ and $3$ .

$\int x^{9} + 3 \cdot 1 x^{2} x^{2} x^{3} + 3 \cdot 1 \cdot 1 x^{2} x^{3} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.20

Multiply $3$ by $1$ .

$\int x^{9} + 3 x^{2} x^{2} x^{3} + 3 \cdot 1 \cdot 1 x^{2} x^{3} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.21

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

$\int x^{9} + 3 x^{2 + 2} x^{3} + 3 \cdot 1 \cdot 1 x^{2} x^{3} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.22

Add $2$ and $2$ .

$\int x^{9} + 3 x^{4} x^{3} + 3 \cdot 1 \cdot 1 x^{2} x^{3} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.23

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

$\int x^{9} + 3 x^{4 + 3} + 3 \cdot 1 \cdot 1 x^{2} x^{3} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.24

Add $4$ and $3$ .

$\int x^{9} + 3 x^{7} + 3 \cdot 1 \cdot 1 x^{2} x^{3} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.25

Multiply $3$ by $1$ .

$\int x^{9} + 3 x^{7} + 3 \cdot 1 x^{2} x^{3} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.26

Multiply $3$ by $1$ .

$\int x^{9} + 3 x^{7} + 3 x^{2} x^{3} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.27

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

$\int x^{9} + 3 x^{7} + 3 x^{2 + 3} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.28

Add $2$ and $3$ .

$\int x^{9} + 3 x^{7} + 3 x^{5} + 1 \cdot 1 \cdot 1 x^{3} d x$

Step 1.29

Multiply $1$ by $1$ .

$\int x^{9} + 3 x^{7} + 3 x^{5} + 1 \cdot 1 x^{3} d x$

Step 1.30

Multiply $1$ by $1$ .

$\int x^{9} + 3 x^{7} + 3 x^{5} + 1 x^{3} d x$

Step 1.31

Multiply $x^{3}$ by $1$ .

$\int x^{9} + 3 x^{7} + 3 x^{5} + x^{3} d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{9} d x + \int 3 x^{7} d x + \int 3 x^{5} d x + \int x^{3} d x$

Step 3

By the Power Rule, the integral of $x^{9}$ with respect to $x$ is $\frac{1}{10} x^{10}$ .

$\frac{1}{10} x^{10} + C + \int 3 x^{7} d x + \int 3 x^{5} d x + \int x^{3} d x$

Step 4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$\frac{1}{10} x^{10} + C + 3 \int x^{7} d x + \int 3 x^{5} d x + \int x^{3} d x$

Step 5

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$\frac{1}{10} x^{10} + C + 3 (\frac{1}{8} x^{8} + C) + \int 3 x^{5} d x + \int x^{3} d x$

Step 6

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$\frac{1}{10} x^{10} + C + 3 (\frac{1}{8} x^{8} + C) + 3 \int x^{5} d x + \int x^{3} d x$

Step 7

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$\frac{1}{10} x^{10} + C + 3 (\frac{1}{8} x^{8} + C) + 3 (\frac{1}{6} x^{6} + C) + \int x^{3} d x$

Step 8

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

$\frac{1}{10} x^{10} + C + 3 (\frac{1}{8} x^{8} + C) + 3 (\frac{1}{6} x^{6} + C) + \frac{1}{4} x^{4} + C$

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{8}$ and $x^{8}$ .

$\frac{1}{10} x^{10} + C + 3 (\frac{x^{8}}{8} + C) + 3 (\frac{1}{6} x^{6} + C) + \frac{1}{4} x^{4} + C$

Step 9.1.2

Combine $\frac{1}{6}$ and $x^{6}$ .

$\frac{1}{10} x^{10} + C + 3 (\frac{x^{8}}{8} + C) + 3 (\frac{x^{6}}{6} + C) + \frac{1}{4} x^{4} + C$

Step 9.2

Simplify.

$\frac{x^{10}}{10} + \frac{3 x^{8}}{8} + \frac{x^{6}}{2} + \frac{1}{4} x^{4} + C$

Step 9.3

Reorder terms.

$\frac{1}{10} x^{10} + \frac{3}{8} x^{8} + \frac{1}{2} x^{6} + \frac{1}{4} x^{4} + C$