Calculus Examples

$f (x) = 7 x^{4} - 6 x^{5} + 5 x^{2} + c$

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.

$F (x) = \int 7 x^{4} - 6 x^{5} + 5 x^{2} + c d x$

Step 3

Split the single integral into multiple integrals.

$\int 7 x^{4} d x + \int - 6 x^{5} d x + \int 5 x^{2} d x + \int c d x$

Step 4

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int x^{4} d x + \int - 6 x^{5} d x + \int 5 x^{2} d x + \int c d x$

Step 5

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

$7 (\frac{1}{5} x^{5} + C) + \int - 6 x^{5} d x + \int 5 x^{2} d x + \int c d x$

Step 6

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

$7 (\frac{1}{5} x^{5} + C) - 6 \int x^{5} d x + \int 5 x^{2} d x + \int c d x$

Step 7

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

$7 (\frac{1}{5} x^{5} + C) - 6 (\frac{1}{6} x^{6} + C) + \int 5 x^{2} d x + \int c d x$

Step 8

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

$7 (\frac{1}{5} x^{5} + C) - 6 (\frac{1}{6} x^{6} + C) + 5 \int x^{2} d x + \int c d x$

Step 9

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

$7 (\frac{1}{5} x^{5} + C) - 6 (\frac{1}{6} x^{6} + C) + 5 (\frac{1}{3} x^{3} + C) + \int c d x$

Step 10

Apply the constant rule.

$7 (\frac{1}{5} x^{5} + C) - 6 (\frac{1}{6} x^{6} + C) + 5 (\frac{1}{3} x^{3} + C) + c x + C$

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

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

$7 (\frac{x^{5}}{5} + C) - 6 (\frac{1}{6} x^{6} + C) + 5 (\frac{1}{3} x^{3} + C) + c x + C$

Step 11.1.2

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

$7 (\frac{x^{5}}{5} + C) - 6 (\frac{x^{6}}{6} + C) + 5 (\frac{1}{3} x^{3} + C) + c x + C$

Step 11.1.3

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

$7 (\frac{x^{5}}{5} + C) - 6 (\frac{x^{6}}{6} + C) + 5 (\frac{x^{3}}{3} + C) + c x + C$

Step 11.2

Simplify.

$\frac{7 x^{5}}{5} - x^{6} + \frac{5 x^{3}}{3} + c x + C$

Step 11.3

Reorder terms.

$\frac{7}{5} x^{5} - x^{6} + \frac{5}{3} x^{3} + c x + C$

Step 12

The answer is the antiderivative of the function $f (x) = 7 x^{4} - 6 x^{5} + 5 x^{2} + c$ .

$F (x) =$ $\frac{7}{5} x^{5} - x^{6} + \frac{5}{3} x^{3} + c x + C$