Calculus Examples

$\int (x + 1) (3 x - 2) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int x (3 x - 2) + 1 (3 x - 2) d x$

Step 1.2

Apply the distributive property.

$\int x (3 x) + x \cdot - 2 + 1 (3 x - 2) d x$

Step 1.3

Apply the distributive property.

$\int x (3 x) + x \cdot - 2 + 1 (3 x) + 1 \cdot - 2 d x$

Step 1.4

Reorder $x$ and $3$ .

$\int 3 \cdot x \cdot x + x \cdot - 2 + 1 (3 x) + 1 \cdot - 2 d x$

Step 1.5

Reorder $x$ and $- 2$ .

$\int 3 \cdot x \cdot x - 2 \cdot x + 1 (3 x) + 1 \cdot - 2 d x$

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Add $1$ and $1$ .

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

Step 1.10

Multiply $3$ by $1$ .

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

Step 1.11

Multiply $- 2$ by $1$ .

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

Step 1.12

Add $- 2 x$ and $3 x$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Reorder terms.

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