Calculus Examples

$\sqrt[3]{x} (x - 4)$

Step 1

Write $\sqrt[3]{x} (x - 4)$ as a function.

$f (x) = \sqrt[3]{x} (x - 4)$

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.

$F (x) = \int \sqrt[3]{x} (x - 4) d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x - 4$ and $d v = \sqrt[3]{x}$ .

$(x - 4) (\frac{3}{4} x^{\frac{4}{3}}) - \int \frac{3}{4} x^{\frac{4}{3}} d x$

Step 5

Simplify.

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Step 5.1

Combine $\frac{3}{4}$ and $x^{\frac{4}{3}}$ .

$(x - 4) \frac{3 x^{\frac{4}{3}}}{4} - \int \frac{3}{4} x^{\frac{4}{3}} d x$

Step 5.2

Combine $\frac{3}{4}$ and $x^{\frac{4}{3}}$ .

$(x - 4) \frac{3 x^{\frac{4}{3}}}{4} - \int \frac{3 x^{\frac{4}{3}}}{4} d x$

Step 6

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

$(x - 4) \frac{3 x^{\frac{4}{3}}}{4} - (\frac{3}{4} \int x^{\frac{4}{3}} d x)$

Step 7

By the Power Rule, the integral of $x^{\frac{4}{3}}$ with respect to $x$ is $\frac{3}{7} x^{\frac{7}{3}}$ .

$(x - 4) \frac{3 x^{\frac{4}{3}}}{4} - \frac{3}{4} (\frac{3}{7} x^{\frac{7}{3}} + C)$

Step 8

Simplify the answer.

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Step 8.1

Rewrite $(x - 4) \frac{3 x^{\frac{4}{3}}}{4} - \frac{3}{4} (\frac{3}{7} x^{\frac{7}{3}} + C)$ as $\frac{3 x^{\frac{7}{3}}}{4} - 3 x^{\frac{4}{3}} - \frac{3}{4} (\frac{3}{7} x^{\frac{7}{3}}) + C$ .

$\frac{3 x^{\frac{7}{3}}}{4} - 3 x^{\frac{4}{3}} - \frac{3}{4} (\frac{3}{7} x^{\frac{7}{3}}) + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $\frac{3}{7}$ and $x^{\frac{7}{3}}$ .

$\frac{3 x^{\frac{7}{3}}}{4} - 3 x^{\frac{4}{3}} - \frac{3}{4} \cdot \frac{3 x^{\frac{7}{3}}}{7} + C$

Step 8.2.2

Multiply $\frac{3 x^{\frac{7}{3}}}{7}$ by $\frac{3}{4}$ .

$\frac{3 x^{\frac{7}{3}}}{4} - 3 x^{\frac{4}{3}} - \frac{3 x^{\frac{7}{3}} \cdot 3}{7 \cdot 4} + C$

Step 8.2.3

Multiply $3$ by $3$ .

$\frac{3 x^{\frac{7}{3}}}{4} - 3 x^{\frac{4}{3}} - \frac{9 x^{\frac{7}{3}}}{7 \cdot 4} + C$

Step 8.2.4

Multiply $7$ by $4$ .

$\frac{3 x^{\frac{7}{3}}}{4} - 3 x^{\frac{4}{3}} - \frac{9 x^{\frac{7}{3}}}{28} + C$

Step 8.2.5

To write $\frac{3 x^{\frac{7}{3}}}{4}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- 3 x^{\frac{4}{3}} + \frac{3 x^{\frac{7}{3}}}{4} \cdot \frac{7}{7} - \frac{9 x^{\frac{7}{3}}}{28} + C$

Step 8.2.6

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.6.1

Multiply $\frac{3 x^{\frac{7}{3}}}{4}$ by $\frac{7}{7}$ .

$- 3 x^{\frac{4}{3}} + \frac{3 x^{\frac{7}{3}} \cdot 7}{4 \cdot 7} - \frac{9 x^{\frac{7}{3}}}{28} + C$

Step 8.2.6.2

Multiply $4$ by $7$ .

$- 3 x^{\frac{4}{3}} + \frac{3 x^{\frac{7}{3}} \cdot 7}{28} - \frac{9 x^{\frac{7}{3}}}{28} + C$

Step 8.2.7

Combine the numerators over the common denominator.

$- 3 x^{\frac{4}{3}} + \frac{3 x^{\frac{7}{3}} \cdot 7 - 9 x^{\frac{7}{3}}}{28} + C$

Step 8.2.8

Multiply $7$ by $3$ .

$- 3 x^{\frac{4}{3}} + \frac{21 x^{\frac{7}{3}} - 9 x^{\frac{7}{3}}}{28} + C$

Step 8.2.9

Subtract $9 x^{\frac{7}{3}}$ from $21 x^{\frac{7}{3}}$ .

$- 3 x^{\frac{4}{3}} + \frac{12 x^{\frac{7}{3}}}{28} + C$

Step 8.2.10

Factor $4$ out of $12 x^{\frac{7}{3}}$ .

$- 3 x^{\frac{4}{3}} + \frac{4 (3 x^{\frac{7}{3}})}{28} + C$

Step 8.2.11

Cancel the common factors.

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Step 8.2.11.1

Factor $4$ out of $28$ .

$- 3 x^{\frac{4}{3}} + \frac{4 (3 x^{\frac{7}{3}})}{4 (7)} + C$

Step 8.2.11.2

Cancel the common factor.

$- 3 x^{\frac{4}{3}} + \frac{4 (3 x^{\frac{7}{3}})}{4 \cdot 7} + C$

Step 8.2.11.3

Rewrite the expression.

$- 3 x^{\frac{4}{3}} + \frac{3 x^{\frac{7}{3}}}{7} + C$

Step 8.3

Reorder terms.

$- 3 x^{\frac{4}{3}} + \frac{3}{7} x^{\frac{7}{3}} + C$

Step 9

The answer is the antiderivative of the function $f (x) = \sqrt[3]{x} (x - 4)$ .

$F (x) =$ $- 3 x^{\frac{4}{3}} + \frac{3}{7} x^{\frac{7}{3}} + C$