Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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Step 4.1

Rewrite ${(x^{2} + 1)}^{2}$ as $(x^{2} + 1) (x^{2} + 1)$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Reorder $x^{2}$ and $1$ .

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

Step 4.6

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

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

Step 4.7

Add $2$ and $2$ .

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

Multiply $1$ by $1$ .

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

Step 4.11

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

$\frac{1}{5} x^{5} + C + \int 2 x^{2} d x + \int d x$

Step 7

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{1}{5} x^{5} + C + 2 \int x^{2} d x + \int d x$

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

Simplify.

$\frac{x^{5}}{5} + \frac{2 x^{3}}{3} + x + C$

Step 10.3

Reorder terms.

$\frac{1}{5} x^{5} + \frac{2}{3} x^{3} + x + C$

Step 11

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

$F (x) =$ $\frac{1}{5} x^{5} + \frac{2}{3} x^{3} + x + C$