Calculus Examples

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

Step 1

Write $\frac{1}{2} + \frac{3}{4} x^{2} - \frac{4}{5} x^{3}$ as a function.

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

Since $\frac{3}{4}$ is constant with respect to $x$ , move $\frac{3}{4}$ out of the integral.

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

Step 7

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

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

Step 8

Since $- \frac{4}{5}$ is constant with respect to $x$ , move $- \frac{4}{5}$ out of the integral.

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify.

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Step 10.2.1

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

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

Step 10.2.2

Multiply $\frac{x^{4}}{4}$ by $\frac{4}{5}$ .

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

Step 10.2.3

Multiply $4$ by $5$ .

$\frac{x}{2} + \frac{x^{3}}{4} - \frac{x^{4} \cdot 4}{20} + C$

Step 10.2.4

Move $4$ to the left of $x^{4}$ .

$\frac{x}{2} + \frac{x^{3}}{4} - \frac{4 \cdot x^{4}}{20} + C$

Step 10.2.5

Cancel the common factor of $4$ and $20$ .

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Step 10.2.5.1

Factor $4$ out of $4 x^{4}$ .

$\frac{x}{2} + \frac{x^{3}}{4} - \frac{4 (x^{4})}{20} + C$

Step 10.2.5.2

Cancel the common factors.

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Step 10.2.5.2.1

Factor $4$ out of $20$ .

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

Step 10.2.5.2.2

Cancel the common factor.

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

Step 10.2.5.2.3

Rewrite the expression.

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

Step 11

Reorder terms.

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

Step 12

The answer is the antiderivative of the function $f (x) = \frac{1}{2} + \frac{3}{4} x^{2} - \frac{4}{5} x^{3}$ .

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