Calculus Examples

$3 x {(2 x + 3)}^{- 0.5}$

Step 1

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

$3 \int x {(2 x + 3)}^{- 0.5} d x$

Step 2

Let $u_{1} = 2 x + 3$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.1

Let $u_{1} = 2 x + 3$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $2 x + 3$ .

$\frac{d}{d x} [2 x + 3]$

Step 2.1.2

By the Sum Rule, the derivative of $2 x + 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$

Step 2.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [3]$

Step 2.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [3]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 + 0$

Step 2.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$3 \int (\frac{u_{1}}{2} - \frac{3}{2}) {u_{1}}^{- 0.5} \frac{1}{2} d u_{1}$

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{2}$ and ${u_{1}}^{- 0.5}$ .

$3 \int (\frac{u_{1}}{2} - \frac{3}{2}) \frac{{u_{1}}^{- 0.5}}{2} d u_{1}$

Step 3.2

Move ${u_{1}}^{- 0.5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 \int (\frac{u_{1}}{2} - \frac{3}{2}) \frac{1}{2 {u_{1}}^{0.5}} d u_{1}$

Step 4

Let $u_{2} = 2 {u_{1}}^{0.5}$ . Then $d u_{2} = \frac{1}{{u_{1}}^{0.5}} d u_{1}$ , so ${u_{1}}^{0.5} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.1

Let $u_{2} = 2 {u_{1}}^{0.5}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 4.1.1

Differentiate $2 {u_{1}}^{0.5}$ .

$\frac{d}{d u_{1}} [2 {u_{1}}^{0.5}]$

Step 4.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 {u_{1}}^{0.5}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [{u_{1}}^{0.5}]$ .

$2 \frac{d}{d u_{1}} [{u_{1}}^{0.5}]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 0.5$ .

$2 (0.5 {u_{1}}^{- 0.5})$

Step 4.1.4

Multiply.

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Step 4.1.4.1

Multiply $0.5$ by $2$ .

$1 {u_{1}}^{- 0.5}$

Step 4.1.4.2

Multiply ${u_{1}}^{- 0.5}$ by $1$ .

${u_{1}}^{- 0.5}$

Step 4.1.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{u_{1}}^{0.5}}$

Step 4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{\frac{{u_{2}}^{2}}{4}}{2}$ as a product.

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

Step 5.2

Multiply $\frac{{u_{2}}^{2}}{4}$ by $\frac{1}{2}$ .

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

Step 5.3

Multiply $4$ by $2$ .

$3 \int (\frac{{u_{2}}^{2}}{8} - \frac{3}{2}) \frac{1}{2} d u_{2}$

Step 6

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

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

Step 7

Simplify.

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Step 7.1

Multiply $- 1$ by $3$ .

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

Step 7.2

Combine $\frac{1}{2}$ and $3$ .

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

Step 8

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

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

Step 9

Simplify.

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Step 9.1

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

$\frac{3}{2 \cdot 2} \int \frac{{u_{2}}^{2}}{4} - 3 d u_{2}$

Step 9.2

Multiply $2$ by $2$ .

$\frac{3}{4} \int \frac{{u_{2}}^{2}}{4} - 3 d u_{2}$

Step 10

Split the single integral into multiple integrals.

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

Step 11

Since $\frac{1}{4}$ is constant with respect to $u_{2}$ , move $\frac{1}{4}$ out of the integral.

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

Step 12

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

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

Step 13

Apply the constant rule.

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

Step 14

Simplify.

$\frac{3}{4} (\frac{{u_{2}}^{3}}{12} - 3 u_{2}) + C$

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

Replace all occurrences of $u_{2}$ with $2 {u_{1}}^{0.5}$ .

$\frac{3}{4} (\frac{{(2 {u_{1}}^{0.5})}^{3}}{12} - 3 (2 {u_{1}}^{0.5})) + C$

Step 15.2

Replace all occurrences of $u_{1}$ with $2 x + 3$ .

$\frac{3}{4} (\frac{{(2 {(2 x + 3)}^{0.5})}^{3}}{12} - 3 (2 {(2 x + 3)}^{0.5})) + C$

Step 16

Simplify.

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Step 16.1

Simplify each term.

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Step 16.1.1

Simplify the numerator.

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Step 16.1.1.1

Apply the product rule to $2 {(2 x + 3)}^{0.5}$ .

$\frac{3}{4} (\frac{2^{3} {({(2 x + 3)}^{0.5})}^{3}}{12} - 3 (2 {(2 x + 3)}^{0.5})) + C$

Step 16.1.1.2

Raise $2$ to the power of $3$ .

$\frac{3}{4} (\frac{8 {({(2 x + 3)}^{0.5})}^{3}}{12} - 3 (2 {(2 x + 3)}^{0.5})) + C$

Step 16.1.1.3

Multiply the exponents in ${({(2 x + 3)}^{0.5})}^{3}$ .

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Step 16.1.1.3.1

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

$\frac{3}{4} (\frac{8 {(2 x + 3)}^{0.5 \cdot 3}}{12} - 3 (2 {(2 x + 3)}^{0.5})) + C$

Step 16.1.1.3.2

Multiply $0.5$ by $3$ .

$\frac{3}{4} (\frac{8 {(2 x + 3)}^{1.5}}{12} - 3 (2 {(2 x + 3)}^{0.5})) + C$

Step 16.1.2

Cancel the common factor of $8$ and $12$ .

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Step 16.1.2.1

Factor $4$ out of $8 {(2 x + 3)}^{1.5}$ .

$\frac{3}{4} (\frac{4 (2 {(2 x + 3)}^{1.5})}{12} - 3 (2 {(2 x + 3)}^{0.5})) + C$

Step 16.1.2.2

Cancel the common factors.

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Step 16.1.2.2.1

Factor $4$ out of $12$ .

$\frac{3}{4} (\frac{4 (2 {(2 x + 3)}^{1.5})}{4 (3)} - 3 (2 {(2 x + 3)}^{0.5})) + C$

Step 16.1.2.2.2

Cancel the common factor.

$\frac{3}{4} (\frac{4 (2 {(2 x + 3)}^{1.5})}{4 \cdot 3} - 3 (2 {(2 x + 3)}^{0.5})) + C$

Step 16.1.2.2.3

Rewrite the expression.

$\frac{3}{4} (\frac{2 {(2 x + 3)}^{1.5}}{3} - 3 (2 {(2 x + 3)}^{0.5})) + C$

Step 16.1.3

Multiply $2$ by $- 3$ .

$\frac{3}{4} (\frac{2 {(2 x + 3)}^{1.5}}{3} - 6 {(2 x + 3)}^{0.5}) + C$

Step 16.2

To write $- 6 {(2 x + 3)}^{0.5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3}{4} (\frac{2 {(2 x + 3)}^{1.5}}{3} - 6 {(2 x + 3)}^{0.5} \cdot \frac{3}{3}) + C$

Step 16.3

Combine $- 6 {(2 x + 3)}^{0.5}$ and $\frac{3}{3}$ .

$\frac{3}{4} (\frac{2 {(2 x + 3)}^{1.5}}{3} + \frac{- 6 {(2 x + 3)}^{0.5} \cdot 3}{3}) + C$

Step 16.4

Combine the numerators over the common denominator.

$\frac{3}{4} \cdot \frac{2 {(2 x + 3)}^{1.5} - 6 {(2 x + 3)}^{0.5} \cdot 3}{3} + C$

Step 16.5

Cancel the common factor of $3$ .

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Step 16.5.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{2 {(2 x + 3)}^{1.5} - 6 {(2 x + 3)}^{0.5} \cdot 3}{3} + C$

Step 16.5.2

Rewrite the expression.

$\frac{1}{4} (2 {(2 x + 3)}^{1.5} - 6 {(2 x + 3)}^{0.5} \cdot 3) + C$

Step 16.6

Multiply $3$ by $- 6$ .

$\frac{1}{4} (2 {(2 x + 3)}^{1.5} - 18 {(2 x + 3)}^{0.5}) + C$

Step 16.7

Apply the distributive property.

$\frac{1}{4} (2 {(2 x + 3)}^{1.5}) + \frac{1}{4} (- 18 {(2 x + 3)}^{0.5}) + C$

Step 16.8

Cancel the common factor of $2$ .

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Step 16.8.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (2 {(2 x + 3)}^{1.5}) + \frac{1}{4} (- 18 {(2 x + 3)}^{0.5}) + C$

Step 16.8.2

Factor $2$ out of $2 {(2 x + 3)}^{1.5}$ .

$\frac{1}{2 (2)} (2 ({(2 x + 3)}^{1.5})) + \frac{1}{4} (- 18 {(2 x + 3)}^{0.5}) + C$

Step 16.8.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 {(2 x + 3)}^{1.5}) + \frac{1}{4} (- 18 {(2 x + 3)}^{0.5}) + C$

Step 16.8.4

Rewrite the expression.

$\frac{1}{2} {(2 x + 3)}^{1.5} + \frac{1}{4} (- 18 {(2 x + 3)}^{0.5}) + C$

Step 16.9

Combine $\frac{1}{2}$ and ${(2 x + 3)}^{1.5}$ .

$\frac{{(2 x + 3)}^{1.5}}{2} + \frac{1}{4} (- 18 {(2 x + 3)}^{0.5}) + C$

Step 16.10

Cancel the common factor of $2$ .

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Step 16.10.1

Factor $2$ out of $4$ .

$\frac{{(2 x + 3)}^{1.5}}{2} + \frac{1}{2 (2)} (- 18 {(2 x + 3)}^{0.5}) + C$

Step 16.10.2

Factor $2$ out of $- 18 {(2 x + 3)}^{0.5}$ .

$\frac{{(2 x + 3)}^{1.5}}{2} + \frac{1}{2 (2)} (2 (- 9 {(2 x + 3)}^{0.5})) + C$

Step 16.10.3

Cancel the common factor.

$\frac{{(2 x + 3)}^{1.5}}{2} + \frac{1}{2 \cdot 2} (2 (- 9 {(2 x + 3)}^{0.5})) + C$

Step 16.10.4

Rewrite the expression.

$\frac{{(2 x + 3)}^{1.5}}{2} + \frac{1}{2} (- 9 {(2 x + 3)}^{0.5}) + C$

Step 16.11

Combine $- 9$ and $\frac{1}{2}$ .

$\frac{{(2 x + 3)}^{1.5}}{2} + \frac{- 9}{2} {(2 x + 3)}^{0.5} + C$

Step 16.12

Combine $\frac{- 9}{2}$ and ${(2 x + 3)}^{0.5}$ .

$\frac{{(2 x + 3)}^{1.5}}{2} + \frac{- 9 {(2 x + 3)}^{0.5}}{2} + C$

Step 16.13

Combine the numerators over the common denominator.

$\frac{{(2 x + 3)}^{1.5} - 9 {(2 x + 3)}^{0.5}}{2} + C$

Step 16.14

Simplify the numerator.

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Step 16.14.1

Factor ${(2 x + 3)}^{0.5}$ out of ${(2 x + 3)}^{1.5} - 9 {(2 x + 3)}^{0.5}$ .

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Step 16.14.1.1

Factor ${(2 x + 3)}^{0.5}$ out of ${(2 x + 3)}^{1.5}$ .

$\frac{{(2 x + 3)}^{0.5} (2 x + 3) - 9 {(2 x + 3)}^{0.5}}{2} + C$

Step 16.14.1.2

Factor ${(2 x + 3)}^{0.5}$ out of $- 9 {(2 x + 3)}^{0.5}$ .

$\frac{{(2 x + 3)}^{0.5} (2 x + 3) + {(2 x + 3)}^{0.5} \cdot - 9}{2} + C$

Step 16.14.1.3

Factor ${(2 x + 3)}^{0.5}$ out of ${(2 x + 3)}^{0.5} (2 x + 3) + {(2 x + 3)}^{0.5} \cdot - 9$ .

$\frac{{(2 x + 3)}^{0.5} (2 x + 3 - 9)}{2} + C$

Step 16.14.2

Subtract $9$ from $3$ .

$\frac{{(2 x + 3)}^{0.5} (2 x - 6)}{2} + C$

Step 16.14.3

Factor $2$ out of $2 x - 6$ .

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Step 16.14.3.1

Factor $2$ out of $2 x$ .

$\frac{{(2 x + 3)}^{0.5} (2 (x) - 6)}{2} + C$

Step 16.14.3.2

Factor $2$ out of $- 6$ .

$\frac{{(2 x + 3)}^{0.5} (2 x + 2 \cdot - 3)}{2} + C$

Step 16.14.3.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$\frac{{(2 x + 3)}^{0.5} (2 (x - 3))}{2} + C$

$\frac{{(2 x + 3)}^{0.5} \cdot 2 (x - 3)}{2} + C$

Step 16.15

Cancel the common factor of $2$ .

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Step 16.15.1

Cancel the common factor.

$\frac{{(2 x + 3)}^{0.5} \cdot 2 (x - 3)}{2} + C$

Step 16.15.2

Divide ${(2 x + 3)}^{0.5} (x - 3)$ by $1$ .

${(2 x + 3)}^{0.5} (x - 3) + C$

Step 16.16

Apply the distributive property.

${(2 x + 3)}^{0.5} x + {(2 x + 3)}^{0.5} \cdot - 3 + C$

Step 16.17

Move $- 3$ to the left of ${(2 x + 3)}^{0.5}$ .

${(2 x + 3)}^{0.5} x - 3 \cdot {(2 x + 3)}^{0.5} + C$

Step 16.18

Reorder factors in ${(2 x + 3)}^{0.5} x - 3 {(2 x + 3)}^{0.5}$ .

$x {(2 x + 3)}^{0.5} - 3 {(2 x + 3)}^{0.5} + C$