Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.2.1

Multiply $x$ by $x^{\frac{1}{2}}$ .

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.2

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Add $2$ and $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the constant rule.

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

Step 9

Simplify.

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

Step 10

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

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