Calculus Examples

$\int \frac{{(2 x - 5)}^{2}}{\sqrt{x}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int \frac{{(2 x - 5)}^{2}}{x^{\frac{1}{2}}} d x$

Step 1.2

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(2 x - 5)}^{2} {(x^{\frac{1}{2}})}^{- 1} d x$

Step 1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 1.3.1

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

$\int {(2 x - 5)}^{2} x^{\frac{1}{2} \cdot - 1} d x$

Step 1.3.2

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

$\int {(2 x - 5)}^{2} x^{\frac{- 1}{2}} d x$

Step 1.3.3

Move the negative in front of the fraction.

$\int {(2 x - 5)}^{2} x^{- \frac{1}{2}} d x$

Step 2

Let $u_{1} = 2 x - 5$ . 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 $u_{1} = 2 x - 5$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $2 x - 5$ .

$\frac{d}{d x} [2 x - 5]$

Step 2.1.2

By the Sum Rule, the derivative of $2 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]$

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]$

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 $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [- 5]$

Step 2.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 5]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ 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 $d u_{1}$ .

$\int {u_{1}}^{2} {(\frac{u_{1}}{2} + \frac{5}{2})}^{- \frac{1}{2}} \frac{1}{2} d u_{1}$

Step 3

Simplify.

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Step 3.1

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

$\int \frac{{u_{1}}^{2}}{2} {(\frac{u_{1}}{2} + \frac{5}{2})}^{- \frac{1}{2}} d u_{1}$

Step 3.2

Combine $\frac{{u_{1}}^{2}}{2}$ and ${(\frac{u_{1}}{2} + \frac{5}{2})}^{- \frac{1}{2}}$ .

$\int \frac{{u_{1}}^{2} {(\frac{u_{1}}{2} + \frac{5}{2})}^{- \frac{1}{2}}}{2} d u_{1}$

Step 3.3

Move ${(\frac{u_{1}}{2} + \frac{5}{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int \frac{{u_{1}}^{2}}{2 {(\frac{u_{1}}{2} + \frac{5}{2})}^{\frac{1}{2}}} d u_{1}$

Step 4

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

$\frac{1}{2} \int \frac{{u_{1}}^{2}}{{(\frac{u_{1}}{2} + \frac{5}{2})}^{\frac{1}{2}}} d u_{1}$

Step 5

Let $u_{2} = \frac{u_{1}}{2} + \frac{5}{2}$ . Then $d u_{2} = \frac{1}{2} d u_{1}$ , so $2 d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 5.1

Let $u_{2} = \frac{u_{1}}{2} + \frac{5}{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 5.1.1

Differentiate $\frac{u_{1}}{2} + \frac{5}{2}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2} + \frac{5}{2}]$

Step 5.1.2

By the Sum Rule, the derivative of $\frac{u_{1}}{2} + \frac{5}{2}$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [\frac{u_{1}}{2}] + \frac{d}{d u_{1}} [\frac{5}{2}]$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2}] + \frac{d}{d u_{1}} [\frac{5}{2}]$

Step 5.1.3

Evaluate $\frac{d}{d u_{1}} [\frac{u_{1}}{2}]$ .

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Step 5.1.3.1

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

$\frac{1}{2} \frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [\frac{5}{2}]$

Step 5.1.3.2

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

$\frac{1}{2} \cdot 1 + \frac{d}{d u_{1}} [\frac{5}{2}]$

Step 5.1.3.3

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

$\frac{1}{2} + \frac{d}{d u_{1}} [\frac{5}{2}]$

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

Since $\frac{5}{2}$ is constant with respect to $u_{1}$ , the derivative of $\frac{5}{2}$ with respect to $u_{1}$ is $0$ .

$\frac{1}{2} + 0$

Step 5.1.4.2

Add $\frac{1}{2}$ and $0$ .

$\frac{1}{2}$

Step 5.2

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

$\frac{1}{2} \int \frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} \cdot \frac{1}{\frac{1}{2}} d u_{2}$

Step 6

Simplify.

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Step 6.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\frac{1}{2} \int \frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} (1 \cdot 2) d u_{2}$

Step 6.2

Multiply $2$ by $1$ .

$\frac{1}{2} \int \frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} \cdot 2 d u_{2}$

Step 6.3

Combine $\frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}}$ and $2$ .

$\frac{1}{2} \int \frac{{(2 u_{2} - 5)}^{2} \cdot 2}{{u_{2}}^{\frac{1}{2}}} d u_{2}$

Step 6.4

Move $2$ to the left of ${(2 u_{2} - 5)}^{2}$ .

$\frac{1}{2} \int \frac{2 {(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} d u_{2}$

Step 7

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

$\frac{1}{2} (2 \int \frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} d u_{2})$

Step 8

Simplify the expression.

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Step 8.1

Simplify.

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Step 8.1.1

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

$\frac{2}{2} \int \frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} d u_{2}$

Step 8.1.2

Cancel the common factor of $2$ .

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Step 8.1.2.1

Cancel the common factor.

$\frac{2}{2} \int \frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} d u_{2}$

Step 8.1.2.2

Rewrite the expression.

$1 \int \frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} d u_{2}$

Step 8.1.3

Multiply $\int \frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} d u_{2}$ by $1$ .

$\int \frac{{(2 u_{2} - 5)}^{2}}{{u_{2}}^{\frac{1}{2}}} d u_{2}$

Step 8.2

Apply basic rules of exponents.

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Step 8.2.1

Move ${u_{2}}^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(2 u_{2} - 5)}^{2} {({u_{2}}^{\frac{1}{2}})}^{- 1} d u_{2}$

Step 8.2.2

Multiply the exponents in ${({u_{2}}^{\frac{1}{2}})}^{- 1}$ .

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Step 8.2.2.1

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

$\int {(2 u_{2} - 5)}^{2} {u_{2}}^{\frac{1}{2} \cdot - 1} d u_{2}$

Step 8.2.2.2

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

$\int {(2 u_{2} - 5)}^{2} {u_{2}}^{\frac{- 1}{2}} d u_{2}$

Step 8.2.2.3

Move the negative in front of the fraction.

$\int {(2 u_{2} - 5)}^{2} {u_{2}}^{- \frac{1}{2}} d u_{2}$

Step 9

Simplify.

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Step 9.1

Rewrite ${(2 u_{2} - 5)}^{2}$ as $(2 u_{2} - 5) (2 u_{2} - 5)$ .

$\int (2 u_{2} - 5) (2 u_{2} - 5) {u_{2}}^{- \frac{1}{2}} d u_{2}$

Step 9.2