Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Move $x^{2}$ .

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

Step 4.6

Move $x^{2}$ .

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

Step 4.7

Multiply $2$ by $2$ .

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

Step 4.8

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

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

Step 4.9

Add $2$ and $2$ .

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

Step 4.10

Multiply $2$ by $- 1$ .

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

Step 4.11

Multiply $- 1$ by $2$ .

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

Step 4.12

Multiply $- 1$ by $- 1$ .

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

Step 4.13

Subtract $2 x^{2}$ from $- 2 x^{2}$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

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

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

Step 11.1.2

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

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

Step 11.2

Simplify.

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

Step 12

Reorder terms.

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

Step 13

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

$F (x) =$ $\frac{4}{5} x^{5} - \frac{4}{3} x^{3} + x + C$