Calculus Examples

$f (x) = \sqrt{x} (6 + 5 x)$

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.

$F (x) = \int \sqrt{x} (6 + 5 x) d x$

Step 3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 6 + 5 x$ and $d v = \sqrt{x}$ .

$(6 + 5 x) (\frac{2}{3} x^{\frac{3}{2}}) - \int \frac{2}{3} x^{\frac{3}{2}} \cdot 5 d x$

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

$(6 + 5 x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2}{3} x^{\frac{3}{2}} \cdot 5 d x$

Step 4.2

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

$(6 + 5 x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2 x^{\frac{3}{2}}}{3} \cdot 5 d x$

Step 4.3

Combine $\frac{2 x^{\frac{3}{2}}}{3}$ and $5$ .

$(6 + 5 x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2 x^{\frac{3}{2}} \cdot 5}{3} d x$

Step 4.4

Multiply $5$ by $2$ .

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

Step 5

Since $\frac{10}{3}$ is constant with respect to $x$ , move $\frac{10}{3}$ out of the integral.

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

Step 6

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

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

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

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

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}}}{3} - \frac{10}{3} (\frac{2}{5} x^{\frac{5}{2}}) + C$

Step 7.2

Simplify.

Tap for more steps...

Step 7.2.1

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

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}}}{3} - \frac{10}{3} \cdot \frac{2 x^{\frac{5}{2}}}{5} + C$

Step 7.2.2

Multiply $\frac{2 x^{\frac{5}{2}}}{5}$ by $\frac{10}{3}$ .

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}}}{3} - \frac{2 x^{\frac{5}{2}} \cdot 10}{5 \cdot 3} + C$

Step 7.2.3

Multiply $10$ by $2$ .

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}}}{3} - \frac{20 x^{\frac{5}{2}}}{5 \cdot 3} + C$

Step 7.2.4

Multiply $5$ by $3$ .

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}}}{3} - \frac{20 x^{\frac{5}{2}}}{15} + C$

Step 7.2.5

Factor $5$ out of $20 x^{\frac{5}{2}}$ .

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}}}{3} - \frac{5 (4 x^{\frac{5}{2}})}{15} + C$

Step 7.2.6

Cancel the common factors.

Tap for more steps...

Step 7.2.6.1

Factor $5$ out of $15$ .

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}}}{3} - \frac{5 (4 x^{\frac{5}{2}})}{5 (3)} + C$

Step 7.2.6.2

Cancel the common factor.

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}}}{3} - \frac{5 (4 x^{\frac{5}{2}})}{5 \cdot 3} + C$

Step 7.2.6.3

Rewrite the expression.

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}}}{3} - \frac{4 x^{\frac{5}{2}}}{3} + C$

Step 7.2.7

Combine the numerators over the common denominator.

$4 x^{\frac{3}{2}} + \frac{10 x^{\frac{5}{2}} - 4 x^{\frac{5}{2}}}{3} + C$

Step 7.2.8

Subtract $4 x^{\frac{5}{2}}$ from $10 x^{\frac{5}{2}}$ .

$4 x^{\frac{3}{2}} + \frac{6 x^{\frac{5}{2}}}{3} + C$

Step 7.2.9

Factor $3$ out of $6 x^{\frac{5}{2}}$ .

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

Step 7.2.10

Cancel the common factors.

Tap for more steps...

Step 7.2.10.1

Factor $3$ out of $3$ .

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

Step 7.2.10.2

Cancel the common factor.

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

Step 7.2.10.3

Rewrite the expression.

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

Step 7.2.10.4

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

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

Step 8

The answer is the antiderivative of the function $f (x) = \sqrt{x} (6 + 5 x)$ .

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