Calculus Examples

$x^{3} - 3 x^{2} - 10 x$

Step 1

Write $x^{3} - 3 x^{2} - 10 x$ as a function.

$f (x) = x^{3} - 3 x^{2} - 10 x$

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.

$F (x) = \int x^{3} - 3 x^{2} - 10 x d x$

Step 4

Split the single integral into multiple integrals.

$\int x^{3} d x + \int - 3 x^{2} d x + \int - 10 x d x$

Step 5

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

$\frac{1}{4} x^{4} + C + \int - 3 x^{2} d x + \int - 10 x d x$

Step 6

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$\frac{1}{4} x^{4} + C - 3 \int x^{2} d x + \int - 10 x d x$

Step 7

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

$\frac{1}{4} x^{4} + C - 3 (\frac{1}{3} x^{3} + C) + \int - 10 x d x$

Step 8

Since $- 10$ is constant with respect to $x$ , move $- 10$ out of the integral.

$\frac{1}{4} x^{4} + C - 3 (\frac{1}{3} x^{3} + C) - 10 \int x d x$

Step 9

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

$\frac{1}{4} x^{4} + C - 3 (\frac{1}{3} x^{3} + C) - 10 (\frac{1}{2} x^{2} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{x^{4}}{4} - x^{3} - 10 (\frac{1}{2} x^{2}) + C$

Step 10.2

Simplify.

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Step 10.2.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{4}}{4} - x^{3} - 10 \frac{x^{2}}{2} + C$

Step 10.2.2

Combine $- 10$ and $\frac{x^{2}}{2}$ .

$\frac{x^{4}}{4} - x^{3} + \frac{- 10 x^{2}}{2} + C$

Step 10.2.3

Cancel the common factor of $- 10$ and $2$ .

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Step 10.2.3.1

Factor $2$ out of $- 10 x^{2}$ .

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

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $2$ out of $2$ .

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

Step 10.2.3.2.2

Cancel the common factor.

$\frac{x^{4}}{4} - x^{3} + \frac{2 (- 5 x^{2})}{2 \cdot 1} + C$

Step 10.2.3.2.3

Rewrite the expression.

$\frac{x^{4}}{4} - x^{3} + \frac{- 5 x^{2}}{1} + C$

Step 10.2.3.2.4

Divide $- 5 x^{2}$ by $1$ .

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

Step 10.3

Reorder terms.

$\frac{1}{4} x^{4} - x^{3} - 5 x^{2} + C$

Step 11

The answer is the antiderivative of the function $f (x) = x^{3} - 3 x^{2} - 10 x$ .

$F (x) =$ $\frac{1}{4} x^{4} - x^{3} - 5 x^{2} + C$