Calculus Examples

$5 x (\sqrt{x} - x^{2})$

Step 1

Write $5 x (\sqrt{x} - x^{2})$ as a function.

$f (x) = 5 x (\sqrt{x} - x^{2})$

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.

$F (x) = \int 5 x (\sqrt{x} - x^{2}) d x$

Step 4

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

$5 \int x (\sqrt{x} - x^{2}) d x$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$5 \int x (x^{\frac{1}{2}} - x^{2}) d x$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Reorder $x$ and $- 1$ .

$5 \int x \cdot x^{\frac{1}{2}} - 1 \cdot x \cdot x^{2} d x$

Step 6.3

Raise $x$ to the power of $1$ .

$5 \int x^{1} x^{\frac{1}{2}} - 1 x \cdot x^{2} d x$

Step 6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.5

Write $1$ as a fraction with a common denominator.

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

Step 6.6

Combine the numerators over the common denominator.

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

Step 6.7

Add $2$ and $1$ .

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

Step 6.8

Factor out negative.

$5 \int x^{\frac{3}{2}} - (x \cdot x^{2}) d x$

Step 6.9

Raise $x$ to the power of $1$ .

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

Step 6.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.11

Add $1$ and $2$ .

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

Step 6.12

Reorder $x^{\frac{3}{2}}$ and $- x^{3}$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 9

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

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

Step 10

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

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

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

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

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

Step 11.1.2

Combine $\frac{2}{5}$ and $x^{\frac{5}{2}}$ .

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

Step 11.2

Simplify.

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

Step 12

The answer is the antiderivative of the function $f (x) = 5 x (\sqrt{x} - x^{2})$ .

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