Calculus Examples

$\int (2 x + 1) (2 x - 1) d x$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Apply the distributive property.

$\int 2 x (2 x - 1) + 1 (2 x - 1) d x$

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Move $x$ .

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

Step 1.5

Move $x$ .

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

Step 1.6

Multiply $2$ by $2$ .

$\int 4 x \cdot x + 2 \cdot - 1 x + 1 \cdot 2 x + 1 \cdot - 1 d x$

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

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

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

Step 1.10

Add $1$ and $1$ .

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

Step 1.11

Multiply $2$ by $- 1$ .

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

Step 1.12

Multiply $2$ by $1$ .

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

Step 1.13

Multiply $- 1$ by $1$ .

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

Step 1.14

Add $- 2 x$ and $2 x$ .

$\int 4 x^{2} + 0 - 1 d x$

Step 1.15

Subtract $1$ from $0$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

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

Step 6.2

Simplify.

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

Step 6.3

Reorder terms.

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