Calculus Examples

$g (x) = x^{6} - 5 x^{4} + 9 x^{2} - 10$

Step 1

The function $G (x)$ can be found by finding the indefinite integral of the derivative $g (x)$ .

$G (x) = \int g (x) d x$

Step 2

Set up the integral to solve.

$G (x) = \int x^{6} - 5 x^{4} + 9 x^{2} - 10 d x$

Step 3

Split the single integral into multiple integrals.

$\int x^{6} d x + \int - 5 x^{4} d x + \int 9 x^{2} d x + \int - 10 d x$

Step 4

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

$\frac{1}{7} x^{7} + C + \int - 5 x^{4} d x + \int 9 x^{2} d x + \int - 10 d x$

Step 5

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$\frac{1}{7} x^{7} + C - 5 \int x^{4} d x + \int 9 x^{2} d x + \int - 10 d x$

Step 6

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

$\frac{1}{7} x^{7} + C - 5 (\frac{1}{5} x^{5} + C) + \int 9 x^{2} d x + \int - 10 d x$

Step 7

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$\frac{1}{7} x^{7} + C - 5 (\frac{1}{5} x^{5} + C) + 9 \int x^{2} d x + \int - 10 d x$

Step 8

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

$\frac{1}{7} x^{7} + C - 5 (\frac{1}{5} x^{5} + C) + 9 (\frac{1}{3} x^{3} + C) + \int - 10 d x$

Step 9

Apply the constant rule.

$\frac{1}{7} x^{7} + C - 5 (\frac{1}{5} x^{5} + C) + 9 (\frac{1}{3} x^{3} + C) - 10 x + C$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$\frac{1}{7} x^{7} + C - 5 (\frac{x^{5}}{5} + C) + 9 (\frac{1}{3} x^{3} + C) - 10 x + C$

Step 10.1.2

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

$\frac{1}{7} x^{7} + C - 5 (\frac{x^{5}}{5} + C) + 9 (\frac{x^{3}}{3} + C) - 10 x + C$

Step 10.2

Simplify.

$\frac{x^{7}}{7} - x^{5} + 3 x^{3} - 10 x + C$

Step 10.3

Reorder terms.

$\frac{1}{7} x^{7} - x^{5} + 3 x^{3} - 10 x + C$

Step 11

The answer is the antiderivative of the function $g (x) = x^{6} - 5 x^{4} + 9 x^{2} - 10$ .

$G (x) =$ $\frac{1}{7} x^{7} - x^{5} + 3 x^{3} - 10 x + C$