Calculus Examples

$f (x) = 6 x^{5} + x^{4} + 2 x^{2} - 3$

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.

$F (x) = \int 6 x^{5} + x^{4} + 2 x^{2} - 3 d x$

Step 3

Split the single integral into multiple integrals.

$\int 6 x^{5} d x + \int x^{4} d x + \int 2 x^{2} d x + \int - 3 d x$

Step 4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int x^{5} d x + \int x^{4} d x + \int 2 x^{2} d x + \int - 3 d x$

Step 5

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$6 (\frac{1}{6} x^{6} + C) + \int x^{4} d x + \int 2 x^{2} d x + \int - 3 d x$

Step 6

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$6 (\frac{1}{6} x^{6} + C) + \frac{1}{5} x^{5} + C + \int 2 x^{2} d x + \int - 3 d x$

Step 7

Combine $\frac{1}{6}$ and $x^{6}$ .

$6 (\frac{x^{6}}{6} + C) + \frac{1}{5} x^{5} + C + \int 2 x^{2} d x + \int - 3 d x$

Step 8

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

$6 (\frac{x^{6}}{6} + C) + \frac{1}{5} x^{5} + C + 2 \int x^{2} d x + \int - 3 d x$

Step 9

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

$6 (\frac{x^{6}}{6} + C) + \frac{1}{5} x^{5} + C + 2 (\frac{1}{3} x^{3} + C) + \int - 3 d x$

Step 10

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

$6 (\frac{x^{6}}{6} + C) + \frac{1}{5} x^{5} + C + 2 (\frac{x^{3}}{3} + C) + \int - 3 d x$

Step 11

Apply the constant rule.

$6 (\frac{x^{6}}{6} + C) + \frac{1}{5} x^{5} + C + 2 (\frac{x^{3}}{3} + C) - 3 x + C$

Step 12

Simplify.

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Step 12.1

Simplify.

$x^{6} + \frac{x^{5}}{5} + \frac{2 x^{3}}{3} - 3 x + C$

Step 12.2

Reorder terms.

$x^{6} + \frac{1}{5} x^{5} + \frac{2}{3} x^{3} - 3 x + C$

Step 13

The answer is the antiderivative of the function $f (x) = 6 x^{5} + x^{4} + 2 x^{2} - 3$ .

$F (x) =$ $x^{6} + \frac{1}{5} x^{5} + \frac{2}{3} x^{3} - 3 x + C$