Calculus Examples

$\sqrt{x} (6 + 10 x)$

Step 1

Write $\sqrt{x} (6 + 10 x)$ as a function.

$f (x) = \sqrt{x} (6 + 10 x)$

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.

$F (x) = \int \sqrt{x} (6 + 10 x) d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 6 + 10 x$ and $d v = \sqrt{x}$ .

$(6 + 10 x) (\frac{2}{3} x^{\frac{3}{2}}) - \int \frac{2}{3} x^{\frac{3}{2}} \cdot 10 d x$

Step 5

Simplify.

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Step 5.1

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

$(6 + 10 x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2}{3} x^{\frac{3}{2}} \cdot 10 d x$

Step 5.2

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

$(6 + 10 x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2 x^{\frac{3}{2}}}{3} \cdot 10 d x$

Step 5.3

Combine $\frac{2 x^{\frac{3}{2}}}{3}$ and $10$ .

$(6 + 10 x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2 x^{\frac{3}{2}} \cdot 10}{3} d x$

Step 5.4

Multiply $10$ by $2$ .

$(6 + 10 x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{20 x^{\frac{3}{2}}}{3} d x$

Step 6

Since $\frac{20}{3}$ is constant with respect to $x$ , move $\frac{20}{3}$ out of the integral.

$(6 + 10 x) \frac{2 x^{\frac{3}{2}}}{3} - (\frac{20}{3} \int x^{\frac{3}{2}} d x)$

Step 7

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

$(6 + 10 x) \frac{2 x^{\frac{3}{2}}}{3} - \frac{20}{3} (\frac{2}{5} x^{\frac{5}{2}} + C)$

Step 8

Simplify the answer.

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Step 8.1

Rewrite $(6 + 10 x) \frac{2 x^{\frac{3}{2}}}{3} - \frac{20}{3} (\frac{2}{5} x^{\frac{5}{2}} + C)$ as $4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{20}{3} (\frac{2}{5} x^{\frac{5}{2}}) + C$ .

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{20}{3} (\frac{2}{5} x^{\frac{5}{2}}) + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{20}{3} \cdot \frac{2 x^{\frac{5}{2}}}{5} + C$

Step 8.2.2

Multiply $\frac{2 x^{\frac{5}{2}}}{5}$ by $\frac{20}{3}$ .

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{2 x^{\frac{5}{2}} \cdot 20}{5 \cdot 3} + C$

Step 8.2.3

Multiply $20$ by $2$ .

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{40 x^{\frac{5}{2}}}{5 \cdot 3} + C$

Step 8.2.4

Multiply $5$ by $3$ .

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{40 x^{\frac{5}{2}}}{15} + C$

Step 8.2.5

Factor $5$ out of $40 x^{\frac{5}{2}}$ .

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{5 (8 x^{\frac{5}{2}})}{15} + C$

Step 8.2.6

Cancel the common factors.

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Step 8.2.6.1

Factor $5$ out of $15$ .

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{5 (8 x^{\frac{5}{2}})}{5 (3)} + C$

Step 8.2.6.2

Cancel the common factor.

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{5 (8 x^{\frac{5}{2}})}{5 \cdot 3} + C$

Step 8.2.6.3

Rewrite the expression.

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}}}{3} - \frac{8 x^{\frac{5}{2}}}{3} + C$

Step 8.2.7

Combine the numerators over the common denominator.

$4 x^{\frac{3}{2}} + \frac{20 x^{\frac{5}{2}} - 8 x^{\frac{5}{2}}}{3} + C$

Step 8.2.8

Subtract $8 x^{\frac{5}{2}}$ from $20 x^{\frac{5}{2}}$ .

$4 x^{\frac{3}{2}} + \frac{12 x^{\frac{5}{2}}}{3} + C$

Step 8.2.9

Factor $3$ out of $12 x^{\frac{5}{2}}$ .

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

Step 8.2.10

Cancel the common factors.

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Step 8.2.10.1

Factor $3$ out of $3$ .

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

Step 8.2.10.2

Cancel the common factor.

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

Step 8.2.10.3

Rewrite the expression.

$4 x^{\frac{3}{2}} + \frac{4 x^{\frac{5}{2}}}{1} + C$

Step 8.2.10.4

Divide $4 x^{\frac{5}{2}}$ by $1$ .

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

Step 9

The answer is the antiderivative of the function $f (x) = \sqrt{x} (6 + 10 x)$ .

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