Calculus Examples

$3 \sqrt{x} (1 - 2 x)$

Step 1

Write $3 \sqrt{x} (1 - 2 x)$ as a function.

$f (x) = 3 \sqrt{x} (1 - 2 x)$

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.

$F (x) = \int 3 \sqrt{x} (1 - 2 x) d x$

Step 4

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

$3 \int \sqrt{x} (1 - 2 x) d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 1 - 2 x$ and $d v = \sqrt{x}$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Multiply $- 2$ by $2$ .

$3 ((1 - 2 x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{- 4 x^{\frac{3}{2}}}{3} d x)$

Step 6.5

Move the negative in front of the fraction.

$3 ((1 - 2 x) \frac{2 x^{\frac{3}{2}}}{3} - \int - \frac{4 x^{\frac{3}{2}}}{3} d x)$

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$3 ((1 - 2 x) \frac{2 x^{\frac{3}{2}}}{3} - - \int \frac{4 x^{\frac{3}{2}}}{3} d x)$

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2

Multiply $\int \frac{4 x^{\frac{3}{2}}}{3} d x$ by $1$ .

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify the answer.

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Step 11.1

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

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

Step 11.2

Rewrite $3 ((1 - 2 x) \frac{2 x^{\frac{3}{2}}}{3} + \frac{4}{3} (\frac{2 x^{\frac{5}{2}}}{5} + C))$ as $3 (\frac{2 x^{\frac{3}{2}}}{3} - \frac{4 x^{\frac{5}{2}}}{3} + \frac{8 x^{\frac{5}{2}}}{15}) + C$ .

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

Step 11.3

Simplify.

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Step 11.3.1

To write $- \frac{4 x^{\frac{5}{2}}}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 11.3.2

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.3.2.1

Multiply $\frac{4 x^{\frac{5}{2}}}{3}$ by $\frac{5}{5}$ .

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

Step 11.3.2.2

Multiply $3$ by $5$ .

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

Step 11.3.3

Combine the numerators over the common denominator.

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

Step 11.3.4

Multiply $5$ by $- 4$ .

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

Step 11.3.5

Add $- 20 x^{\frac{5}{2}}$ and $8 x^{\frac{5}{2}}$ .

$3 (\frac{2 x^{\frac{3}{2}}}{3} + \frac{- 12 x^{\frac{5}{2}}}{15}) + C$

Step 11.3.6

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

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

Step 11.3.7

Cancel the common factors.

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Step 11.3.7.1

Factor $3$ out of $15$ .

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

Step 11.3.7.2

Cancel the common factor.

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

Step 11.3.7.3

Rewrite the expression.

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

Step 11.3.8

Move the negative in front of the fraction.

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

Step 11.4

Simplify.

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Step 11.4.1

Apply the distributive property.

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

Step 11.4.2

Cancel the common factor of $3$ .

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Step 11.4.2.1

Cancel the common factor.

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

Step 11.4.2.2

Rewrite the expression.

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

Step 11.4.3

Multiply $3 (- \frac{4 x^{\frac{5}{2}}}{5})$ .

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Step 11.4.3.1

Multiply $- 1$ by $3$ .

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

Step 11.4.3.2

Combine $- 3$ and $\frac{4 x^{\frac{5}{2}}}{5}$ .

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

Step 11.4.3.3

Multiply $4$ by $- 3$ .

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

Step 11.4.4

Move the negative in front of the fraction.

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

Step 11.5

Reorder terms.

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

Step 12

The answer is the antiderivative of the function $f (x) = 3 \sqrt{x} (1 - 2 x)$ .

$F (x) =$ $2 x^{\frac{3}{2}} - \frac{12}{5} x^{\frac{5}{2}} + C$