Calculus Examples

$(x + 1) (x + 2)$

Step 1

Write $(x + 1) (x + 2)$ as a function.

$f (x) = (x + 1) (x + 2)$

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.

$F (x) = \int (x + 1) (x + 2) d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Reorder $x$ and $2$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Add $1$ and $1$ .

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

Step 4.9

Multiply $x$ by $1$ .

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

Step 4.10

Multiply $2$ by $1$ .

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

Step 4.11

Add $2 x$ and $x$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 10.2

Simplify.

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

Step 10.3

Reorder terms.

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

Step 11

The answer is the antiderivative of the function $f (x) = (x + 1) (x + 2)$ .

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