Calculus Examples

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

Step 1

Write ${(x^{2} + 1)}^{3}$ as a function.

$f (x) = {(x^{2} + 1)}^{3}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

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

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Step 4.1

Use the Binomial Theorem.

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

Step 4.2

Rewrite the exponentiation as a product.

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

Step 4.3

Rewrite the exponentiation as a product.

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

Step 4.4

Rewrite the exponentiation as a product.

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

Step 4.5

Rewrite the exponentiation as a product.

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

Step 4.6

Rewrite the exponentiation as a product.

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

Step 4.7

Rewrite the exponentiation as a product.

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

Step 4.8

Move $x^{2}$ .

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

Step 4.9

Move $x^{2}$ .

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

Step 4.10

Move $x^{2}$ .

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

Step 4.11

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

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

Step 4.12

Add $2$ and $2$ .

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

Step 4.13

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

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

Step 4.14

Add $4$ and $2$ .

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

Step 4.15

Multiply $3$ by $1$ .

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

Step 4.16

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

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

Step 4.17

Add $2$ and $2$ .

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

Step 4.18

Multiply $3$ by $1$ .

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

Step 4.19

Multiply $3$ by $1$ .

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

Step 4.20

Multiply $1$ by $1$ .

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

Step 4.21

Multiply $1$ by $1$ .

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$\frac{1}{7} x^{7} + C + \int 3 x^{4} d x + \int 3 x^{2} d x + \int d x$

Step 7

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

$\frac{1}{7} x^{7} + C + 3 \int x^{4} d x + \int 3 x^{2} d x + \int d x$

Step 8

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

$\frac{1}{7} x^{7} + C + 3 (\frac{1}{5} x^{5} + C) + \int 3 x^{2} d x + \int d x$

Step 9

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

$\frac{1}{7} x^{7} + C + 3 (\frac{1}{5} x^{5} + C) + 3 \int x^{2} d x + \int d x$

Step 10

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

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

Step 11

Apply the constant rule.

$\frac{1}{7} x^{7} + C + 3 (\frac{1}{5} x^{5} + C) + 3 (\frac{1}{3} x^{3} + C) + x + C$

Step 12

Simplify.

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Step 12.1

Simplify.

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Step 12.1.1

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

$\frac{1}{7} x^{7} + C + 3 (\frac{x^{5}}{5} + C) + 3 (\frac{1}{3} x^{3} + C) + x + C$

Step 12.1.2

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

$\frac{1}{7} x^{7} + C + 3 (\frac{x^{5}}{5} + C) + 3 (\frac{x^{3}}{3} + C) + x + C$

$\frac{1}{7} x^{7} + C + 3 (\frac{x^{5}}{5} + C) + 3 (\frac{x^{3}}{3} + C) + x + C$

Step 12.2

Simplify.

$\frac{x^{7}}{7} + \frac{3 x^{5}}{5} + x^{3} + x + C$

Step 12.3

Reorder terms.

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

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

Step 13

The answer is the antiderivative of the function $f (x) = {(x^{2} + 1)}^{3}$ .

$F (x) =$ $\frac{1}{7} x^{7} + \frac{3}{5} x^{5} + x^{3} + x + C$

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