Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

$\int x \cdot x + x \cdot - 2 + 3 x + 3 \cdot - 2 d x$

Step 1.4

Reorder $x$ and $- 2$ .

$\int x \cdot x - 2 \cdot x + 3 x + 3 \cdot - 2 d x$

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

Add $1$ and $1$ .

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

Step 1.9

Multiply $3$ by $- 2$ .

$\int x^{2} - 2 x + 3 x - 6 d x$

Step 1.10

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

$\int x^{2} + x - 6 d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{2} d x + \int x d x + \int - 6 d x$

Step 3

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

$\frac{1}{3} x^{3} + C + \int x d x + \int - 6 d x$

Step 4

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

$\frac{1}{3} x^{3} + C + \frac{1}{2} x^{2} + C + \int - 6 d x$

Step 5

Apply the constant rule.

$\frac{1}{3} x^{3} + C + \frac{1}{2} x^{2} + C - 6 x + C$

Step 6

Simplify.

$\frac{1}{3} x^{3} + \frac{1}{2} x^{2} - 6 x + C$