Calculus Examples

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

Step 1

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

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Rewrite the exponentiation as a product.

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

Step 1.3

Rewrite the exponentiation as a product.

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

Step 1.4

Rewrite the exponentiation as a product.

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Apply the distributive property.

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

Step 1.7

Apply the distributive property.

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

Step 1.8

Move $x^{2}$ .

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

Step 1.9

Move parentheses.

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

Step 1.10

Move $x^{2}$ .

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

Step 1.11

Move $x$ .

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

Step 1.12

Move parentheses.

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

Step 1.13

Move parentheses.

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

Step 1.14

Move $x^{2}$ .

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

Step 1.15

Move parentheses.

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

Step 1.16

Move $x^{2}$ .

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

Step 1.17

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

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

Step 1.18

Add $2$ and $3$ .

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

Step 1.19

Multiply $3$ by $1$ .

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

Step 1.20

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

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

Step 1.21

Add $2$ and $2$ .

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

Step 1.22

Multiply $3$ by $1$ .

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

Step 1.23

Multiply $3$ by $1$ .

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

Step 1.24

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

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

Step 1.25

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

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

Step 1.26

Add $2$ and $1$ .

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

Step 1.27

Multiply $1$ by $1$ .

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

Step 1.28

Multiply $1$ by $1$ .

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

Step 1.29

Multiply $x^{2}$ by $1$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

$\frac{1}{6} x^{6} + C + \int 3 x^{4} d x + \int 3 x^{3} d x + \int x^{2} d x$

Step 4

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

$\frac{1}{6} x^{6} + C + 3 \int x^{4} d x + \int 3 x^{3} d x + \int x^{2} d x$

Step 5

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

$\frac{1}{6} x^{6} + C + 3 (\frac{1}{5} x^{5} + C) + \int 3 x^{3} d x + \int x^{2} d x$

Step 6

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

$\frac{1}{6} x^{6} + C + 3 (\frac{1}{5} x^{5} + C) + 3 \int x^{3} d x + \int x^{2} d x$

Step 7

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

$\frac{1}{6} x^{6} + C + 3 (\frac{1}{5} x^{5} + C) + 3 (\frac{1}{4} x^{4} + C) + \int x^{2} d x$

Step 8

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

$\frac{1}{6} x^{6} + C + 3 (\frac{1}{5} x^{5} + C) + 3 (\frac{1}{4} x^{4} + C) + \frac{1}{3} x^{3} + C$

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

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

$\frac{1}{6} x^{6} + C + 3 (\frac{x^{5}}{5} + C) + 3 (\frac{1}{4} x^{4} + C) + \frac{1}{3} x^{3} + C$

Step 9.1.2

Combine $\frac{1}{4}$ and $x^{4}$ .

$\frac{1}{6} x^{6} + C + 3 (\frac{x^{5}}{5} + C) + 3 (\frac{x^{4}}{4} + C) + \frac{1}{3} x^{3} + C$

Step 9.2

Simplify.

$\frac{x^{6}}{6} + \frac{3 x^{5}}{5} + \frac{3 x^{4}}{4} + \frac{1}{3} x^{3} + C$

Step 9.3

Reorder terms.

$\frac{1}{6} x^{6} + \frac{3}{5} x^{5} + \frac{3}{4} x^{4} + \frac{1}{3} x^{3} + C$