Calculus Examples

$\frac{x + \sqrt{x} + 1}{\sqrt[3]{x}}$

Step 1

Write $\frac{x + \sqrt{x} + 1}{\sqrt[3]{x}}$ as a function.

$f (x) = \frac{x + \sqrt{x} + 1}{\sqrt[3]{x}}$

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.

$F (x) = \int \frac{x + \sqrt{x} + 1}{\sqrt[3]{x}} d x$

Step 4

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

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

Step 5

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

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

Step 6

Move $x^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7

Multiply the exponents in ${(x^{\frac{1}{3}})}^{- 1}$ .

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Step 7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.2

Combine $\frac{1}{3}$ and $- 1$ .

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

Step 7.3

Move the negative in front of the fraction.

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

Combine the numerators over the common denominator.

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

Step 8.7

Subtract $1$ from $3$ .

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

Step 8.8

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

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

Step 8.9

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 8.10

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.11

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.11.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Step 8.11.2

Multiply $2$ by $3$ .

$\int x^{\frac{2}{3}} + x^{\frac{3}{6} - \frac{1}{3} \cdot \frac{2}{2}} + 1 x^{- \frac{1}{3}} d x$

Step 8.11.3

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$\int x^{\frac{2}{3}} + x^{\frac{3}{6} - \frac{2}{3 \cdot 2}} + 1 x^{- \frac{1}{3}} d x$

Step 8.11.4

Multiply $3$ by $2$ .

$\int x^{\frac{2}{3}} + x^{\frac{3}{6} - \frac{2}{6}} + 1 x^{- \frac{1}{3}} d x$

Step 8.12

Combine the numerators over the common denominator.

$\int x^{\frac{2}{3}} + x^{\frac{3 - 2}{6}} + 1 x^{- \frac{1}{3}} d x$

Step 8.13

Subtract $2$ from $3$ .

$\int x^{\frac{2}{3}} + x^{\frac{1}{6}} + 1 x^{- \frac{1}{3}} d x$

Step 8.14

Multiply $x^{- \frac{1}{3}}$ by $1$ .

$\int x^{\frac{2}{3}} + x^{\frac{1}{6}} + x^{- \frac{1}{3}} d x$

Step 9

Split the single integral into multiple integrals.

$\int x^{\frac{2}{3}} d x + \int x^{\frac{1}{6}} d x + \int x^{- \frac{1}{3}} d x$

Step 10

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

$\frac{3}{5} x^{\frac{5}{3}} + C + \int x^{\frac{1}{6}} d x + \int x^{- \frac{1}{3}} d x$

Step 11

By the Power Rule, the integral of $x^{\frac{1}{6}}$ with respect to $x$ is $\frac{6}{7} x^{\frac{7}{6}}$ .

$\frac{3}{5} x^{\frac{5}{3}} + C + \frac{6}{7} x^{\frac{7}{6}} + C + \int x^{- \frac{1}{3}} d x$

Step 12

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

$\frac{3}{5} x^{\frac{5}{3}} + C + \frac{6}{7} x^{\frac{7}{6}} + C + \frac{3}{2} x^{\frac{2}{3}} + C$

Step 13

Simplify.

$\frac{3}{5} x^{\frac{5}{3}} + \frac{6}{7} x^{\frac{7}{6}} + \frac{3}{2} x^{\frac{2}{3}} + C$

Step 14

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

$F (x) =$ $\frac{3}{5} x^{\frac{5}{3}} + \frac{6}{7} x^{\frac{7}{6}} + \frac{3}{2} x^{\frac{2}{3}} + C$