Calculus Examples

$f (x) = 4 + 2 x^{6} + x^{2}$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int 4 + 2 x^{6} + x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int 4 d x + \int 2 x^{6} d x + \int x^{2} d x$

Step 4

Apply the constant rule.

$4 x + C + \int 2 x^{6} d x + \int x^{2} d x$

Step 5

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

$4 x + C + 2 \int x^{6} d x + \int x^{2} d x$

Step 6

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$4 x + C + 2 (\frac{1}{7} x^{7} + C) + \int x^{2} d x$

Step 7

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

$4 x + C + 2 (\frac{1}{7} x^{7} + C) + \frac{1}{3} x^{3} + C$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{7}$ and $x^{7}$ .

$4 x + C + 2 (\frac{x^{7}}{7} + C) + \frac{1}{3} x^{3} + C$

Step 8.2

Simplify.

$4 x + \frac{2 x^{7}}{7} + \frac{1}{3} x^{3} + C$

Step 8.3

Reorder terms.

$4 x + \frac{2}{7} x^{7} + \frac{1}{3} x^{3} + C$

Step 9

The answer is the antiderivative of the function $f (x) = 4 + 2 x^{6} + x^{2}$ .

$F (x) =$ $4 x + \frac{2}{7} x^{7} + \frac{1}{3} x^{3} + C$