Calculus Examples

$\int (4 x - 1) (3 x + 1) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int 4 x (3 x + 1) - 1 (3 x + 1) d x$

Step 1.2

Apply the distributive property.

$\int 4 x (3 x) + 4 x \cdot 1 - 1 (3 x + 1) d x$

Step 1.3

Apply the distributive property.

$\int 4 x (3 x) + 4 x \cdot 1 - 1 (3 x) - 1 \cdot 1 d x$

Step 1.4

Move $x$ .

$\int 4 \cdot 3 x \cdot x + 4 x \cdot 1 - 1 (3 x) - 1 \cdot 1 d x$

Step 1.5

Move $x$ .

$\int 4 \cdot 3 x \cdot x + 4 \cdot 1 x - 1 (3 x) - 1 \cdot 1 d x$

Step 1.6

Multiply $4$ by $3$ .

$\int 12 x \cdot x + 4 \cdot 1 x - 1 \cdot 3 x - 1 \cdot 1 d x$

Step 1.7

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

$\int 12 (x^{1} x) + 4 \cdot 1 x - 1 \cdot 3 x - 1 \cdot 1 d x$

Step 1.8

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

$\int 12 (x^{1} x^{1}) + 4 \cdot 1 x - 1 \cdot 3 x - 1 \cdot 1 d x$

Step 1.9

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

$\int 12 x^{1 + 1} + 4 \cdot 1 x - 1 \cdot 3 x - 1 \cdot 1 d x$

Step 1.10

Add $1$ and $1$ .

$\int 12 x^{2} + 4 \cdot 1 x - 1 \cdot 3 x - 1 \cdot 1 d x$

Step 1.11

Multiply $4$ by $1$ .

$\int 12 x^{2} + 4 x - 1 \cdot 3 x - 1 \cdot 1 d x$

Step 1.12

Multiply $- 1$ by $3$ .

$\int 12 x^{2} + 4 x - 3 x - 1 \cdot 1 d x$

Step 1.13

Multiply $- 1$ by $1$ .

$\int 12 x^{2} + 4 x - 3 x - 1 d x$

Step 1.14

Subtract $3 x$ from $4 x$ .

$\int 12 x^{2} + x - 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int 12 x^{2} d x + \int x d x + \int - 1 d x$

Step 3

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int x^{2} d x + \int x d x + \int - 1 d x$

Step 4

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

$12 (\frac{1}{3} x^{3} + C) + \int x d x + \int - 1 d x$

Step 5

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

$12 (\frac{1}{3} x^{3} + C) + \frac{1}{2} x^{2} + C + \int - 1 d x$

Step 6

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

$12 (\frac{x^{3}}{3} + C) + \frac{1}{2} x^{2} + C + \int - 1 d x$

Step 7

Apply the constant rule.

$12 (\frac{x^{3}}{3} + C) + \frac{1}{2} x^{2} + C - x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

$4 x^{3} + \frac{x^{2}}{2} - x + C$

Step 8.2

Reorder terms.

$4 x^{3} + \frac{1}{2} x^{2} - x + C$