Calculus Examples

$\int_{- 4}^{2} [f (x) + 2 x + 3] d x$

Step 1

Remove parentheses.

$\int_{- 4}^{2} f (x) + 2 x + 3 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 4}^{2} f (x) d x + \int_{- 4}^{2} 2 x d x + \int_{- 4}^{2} 3 d x$

Step 3

Apply the constant rule.

$f (x) x]_{- 4}^{2} + \int_{- 4}^{2} 2 x d x + \int_{- 4}^{2} 3 d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$f (x) x]_{- 4}^{2} + 2 \int_{- 4}^{2} x d x + \int_{- 4}^{2} 3 d x$

Step 5

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

$f (x) x]_{- 4}^{2} + 2 (\frac{1}{2} x^{2}]_{- 4}^{2}) + \int_{- 4}^{2} 3 d x$

Step 6

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

$f (x) x]_{- 4}^{2} + 2 (\frac{x^{2}}{2}]_{- 4}^{2}) + \int_{- 4}^{2} 3 d x$

Step 7

Apply the constant rule.

$f (x) x]_{- 4}^{2} + 2 (\frac{x^{2}}{2}]_{- 4}^{2}) + 3 x]_{- 4}^{2}$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Combine $f (x) x]_{- 4}^{2}$ and $3 x]_{- 4}^{2}$ .

$2 (\frac{x^{2}}{2}]_{- 4}^{2}) + f (x) x + 3 x]_{- 4}^{2}$

Step 8.2

Substitute and simplify.

Tap for more steps...

Step 8.2.1

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $- 4$ .

$2 ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2}) + f (x) x + 3 x]_{- 4}^{2}$

Step 8.2.2

Evaluate $f (x) x + 3 x$ at $2$ and at $- 4$ .

$2 ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3

Simplify.

Tap for more steps...

Step 8.2.3.1

Raise $2$ to the power of $2$ .

$2 (\frac{4}{2} - \frac{{(- 4)}^{2}}{2}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.2

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 8.2.3.2.1

Factor $2$ out of $4$ .

$2 (\frac{2 \cdot 2}{2} - \frac{{(- 4)}^{2}}{2}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 8.2.3.2.2.1

Factor $2$ out of $2$ .

$2 (\frac{2 \cdot 2}{2 (1)} - \frac{{(- 4)}^{2}}{2}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.2.2.2

Cancel the common factor.

$2 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.2.2.3

Rewrite the expression.

$2 (\frac{2}{1} - \frac{{(- 4)}^{2}}{2}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.2.2.4

Divide $2$ by $1$ .

$2 (2 - \frac{{(- 4)}^{2}}{2}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.3

Raise $- 4$ to the power of $2$ .

$2 (2 - \frac{16}{2}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.4

Cancel the common factor of $16$ and $2$ .

Tap for more steps...

Step 8.2.3.4.1

Factor $2$ out of $16$ .

$2 (2 - \frac{2 \cdot 8}{2}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.4.2

Cancel the common factors.

Tap for more steps...

Step 8.2.3.4.2.1

Factor $2$ out of $2$ .

$2 (2 - \frac{2 \cdot 8}{2 (1)}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.4.2.2

Cancel the common factor.

$2 (2 - \frac{2 \cdot 8}{2 \cdot 1}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.4.2.3

Rewrite the expression.

$2 (2 - \frac{8}{1}) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.4.2.4

Divide $8$ by $1$ .

$2 (2 - 1 \cdot 8) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.5

Multiply $- 1$ by $8$ .

$2 (2 - 8) + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.6

Subtract $8$ from $2$ .

$2 \cdot - 6 + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.7

Multiply $2$ by $- 6$ .

$- 12 + (f (x) \cdot 2 + 3 \cdot 2) - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.8

Move $2$ to the left of $f (x)$ .

$- 12 + 2 \cdot f (x) + 3 \cdot 2 - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.9

Multiply $3$ by $2$ .

$- 12 + 2 f (x) + 6 - (f (x) \cdot - 4 + 3 \cdot - 4)$

Step 8.2.3.10

Move $- 4$ to the left of $f (x)$ .

$- 12 + 2 f (x) + 6 - (- 4 \cdot f (x) + 3 \cdot - 4)$

Step 8.2.3.11

Multiply $3$ by $- 4$ .

$- 12 + 2 f (x) + 6 - (- 4 f (x) - 12)$

Step 8.2.3.12

Add $- 12$ and $6$ .

$2 f (x) - 6 - (- 4 f (x) - 12)$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Apply the distributive property.

$2 f (x) - 6 - (- 4 f (x)) - - 12$

Step 9.1.2

Multiply $- 4$ by $- 1$ .

$2 f (x) - 6 + 4 f (x) - - 12$

Step 9.1.3

Multiply $- 1$ by $- 12$ .

$2 f (x) - 6 + 4 f (x) + 12$

Step 9.2

Add $2 f (x)$ and $4 f (x)$ .

$6 f (x) - 6 + 12$

Step 9.3

Add $- 6$ and $12$ .

$6 f (x) + 6$

Step 10