Calculus Examples

$f (x) = 3 \sqrt{x} - 2 \sqrt[3]{x}$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int 3 \sqrt{x} - 2 \sqrt[3]{x} d x$

Step 3

Split the single integral into multiple integrals.

$\int 3 \sqrt{x} d x + \int - 2 \sqrt[3]{x} d x$

Step 4

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

$3 \int \sqrt{x} d x + \int - 2 \sqrt[3]{x} d x$

Step 5

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

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

Step 6

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

$3 (\frac{2}{3} x^{\frac{3}{2}} + C) + \int - 2 \sqrt[3]{x} d x$

Step 7

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$3 (\frac{2}{3} x^{\frac{3}{2}} + C) - 2 \int \sqrt[3]{x} d x$

Step 8

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify.

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Step 10.2.1

Combine $3$ and $\frac{2}{3}$ .

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

Step 10.2.2

Multiply $3$ by $2$ .

$\frac{6}{3} x^{\frac{3}{2}} - 2 (\frac{3}{4}) x^{\frac{4}{3}} + C$

Step 10.2.3

Cancel the common factor of $6$ and $3$ .

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Step 10.2.3.1

Factor $3$ out of $6$ .

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

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $3$ out of $3$ .

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

Step 10.2.3.2.2

Cancel the common factor.

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

Step 10.2.3.2.3

Rewrite the expression.

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

Step 10.2.3.2.4

Divide $2$ by $1$ .

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

Step 10.2.4

Combine $- 2$ and $\frac{3}{4}$ .

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

Step 10.2.5

Multiply $- 2$ by $3$ .

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

Step 10.2.6

Cancel the common factor of $- 6$ and $4$ .

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Step 10.2.6.1

Factor $2$ out of $- 6$ .

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

Step 10.2.6.2

Cancel the common factors.

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Step 10.2.6.2.1

Factor $2$ out of $4$ .

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

Step 10.2.6.2.2

Cancel the common factor.

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

Step 10.2.6.2.3

Rewrite the expression.

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

Step 10.2.7

Move the negative in front of the fraction.

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

Step 11

The answer is the antiderivative of the function $f (x) = 3 \sqrt{x} - 2 \sqrt[3]{x}$ .

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