Calculus Examples

$7 x^{4} - 6 x^{3} + 4 x^{2} + C$

Step 1

Write $7 x^{4} - 6 x^{3} + 4 x^{2} + C$ as a function.

$f (x) = 7 x^{4} - 6 x^{3} + 4 x^{2} + C$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int 7 x^{4} - 6 x^{3} + 4 x^{2} + C d x$

Step 4

Split the single integral into multiple integrals.

$\int 7 x^{4} d x + \int - 6 x^{3} d x + \int 4 x^{2} d x + \int C d x$

Step 5

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

$7 \int x^{4} d x + \int - 6 x^{3} d x + \int 4 x^{2} d x + \int C d x$

Step 6

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

$7 (\frac{1}{5} x^{5} + C) + \int - 6 x^{3} d x + \int 4 x^{2} d x + \int C d x$

Step 7

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

$7 (\frac{1}{5} x^{5} + C) - 6 \int x^{3} d x + \int 4 x^{2} d x + \int C d x$

Step 8

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$7 (\frac{1}{5} x^{5} + C) - 6 (\frac{1}{4} x^{4} + C) + \int 4 x^{2} d x + \int C d x$

Step 9

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$7 (\frac{1}{5} x^{5} + C) - 6 (\frac{1}{4} x^{4} + C) + 4 \int x^{2} d x + \int C d x$

Step 10

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

$7 (\frac{1}{5} x^{5} + C) - 6 (\frac{1}{4} x^{4} + C) + 4 (\frac{1}{3} x^{3} + C) + \int C d x$

Step 11

Apply the constant rule.

$7 (\frac{1}{5} x^{5} + C) - 6 (\frac{1}{4} x^{4} + C) + 4 (\frac{1}{3} x^{3} + C) + C x + C$

Step 12

Simplify.

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Step 12.1

Simplify.

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Step 12.1.1

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

$7 (\frac{x^{5}}{5} + C) - 6 (\frac{1}{4} x^{4} + C) + 4 (\frac{1}{3} x^{3} + C) + C x + C$

Step 12.1.2

Combine $\frac{1}{4}$ and $x^{4}$ .

$7 (\frac{x^{5}}{5} + C) - 6 (\frac{x^{4}}{4} + C) + 4 (\frac{1}{3} x^{3} + C) + C x + C$

Step 12.1.3

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

$7 (\frac{x^{5}}{5} + C) - 6 (\frac{x^{4}}{4} + C) + 4 (\frac{x^{3}}{3} + C) + C x + C$

Step 12.2

Simplify.

$\frac{7 x^{5}}{5} - \frac{3 x^{4}}{2} + \frac{4 x^{3}}{3} + C x + C$

Step 13

Reorder terms.

$\frac{7}{5} x^{5} - \frac{3}{2} x^{4} + \frac{4}{3} x^{3} + C x + C$

Step 14

The answer is the antiderivative of the function $f (x) = 7 x^{4} - 6 x^{3} + 4 x^{2} + C$ .

$F (x) =$ $\frac{7}{5} x^{5} - \frac{3}{2} x^{4} + \frac{4}{3} x^{3} + C x + C$