Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Rewrite ${(\sqrt{a} - \sqrt{x})}^{2}$ as $(\sqrt{a} - \sqrt{x}) (\sqrt{a} - \sqrt{x})$ .

$\int (\sqrt{a} - \sqrt{x}) (\sqrt{a} - \sqrt{x}) d x$

Step 1.2

Expand $(\sqrt{a} - \sqrt{x}) (\sqrt{a} - \sqrt{x})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\int \sqrt{a} (\sqrt{a} - \sqrt{x}) - \sqrt{x} (\sqrt{a} - \sqrt{x}) d x$

Step 1.2.2

Apply the distributive property.

$\int \sqrt{a} \sqrt{a} + \sqrt{a} (- \sqrt{x}) - \sqrt{x} (\sqrt{a} - \sqrt{x}) d x$

Step 1.2.3

Apply the distributive property.

$\int \sqrt{a} \sqrt{a} + \sqrt{a} (- \sqrt{x}) - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $\sqrt{a} \sqrt{a}$ .

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Step 1.3.1.1.1

Raise $\sqrt{a}$ to the power of $1$ .

$\int {\sqrt{a}}^{1} \sqrt{a} + \sqrt{a} (- \sqrt{x}) - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.1.2

Raise $\sqrt{a}$ to the power of $1$ .

$\int {\sqrt{a}}^{1} {\sqrt{a}}^{1} + \sqrt{a} (- \sqrt{x}) - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int {\sqrt{a}}^{1 + 1} + \sqrt{a} (- \sqrt{x}) - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.1.4

Add $1$ and $1$ .

$\int {\sqrt{a}}^{2} + \sqrt{a} (- \sqrt{x}) - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.2

Rewrite ${\sqrt{a}}^{2}$ as $a$ .

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Step 1.3.1.2.1

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

$\int {(a^{\frac{1}{2}})}^{2} + \sqrt{a} (- \sqrt{x}) - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.2.2

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

$\int a^{\frac{1}{2} \cdot 2} + \sqrt{a} (- \sqrt{x}) - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.2.3

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

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

Step 1.3.1.2.4

Cancel the common factor of $2$ .

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Step 1.3.1.2.4.1

Cancel the common factor.

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

Step 1.3.1.2.4.2

Rewrite the expression.

$\int a^{1} + \sqrt{a} (- \sqrt{x}) - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.2.5

Simplify.

$\int a + \sqrt{a} (- \sqrt{x}) - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.3

Rewrite using the commutative property of multiplication.

$\int a - \sqrt{a} \sqrt{x} - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.4

Combine using the product rule for radicals.

$\int a - \sqrt{x a} - \sqrt{x} \sqrt{a} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.5

Combine using the product rule for radicals.

$\int a - \sqrt{x a} - \sqrt{a x} - \sqrt{x} (- \sqrt{x}) d x$

Step 1.3.1.6

Multiply $- \sqrt{x} (- \sqrt{x})$ .

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Step 1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$\int a - \sqrt{x a} - \sqrt{a x} + 1 \sqrt{x} \sqrt{x} d x$

Step 1.3.1.6.2

Multiply $\sqrt{x}$ by $1$ .

$\int a - \sqrt{x a} - \sqrt{a x} + \sqrt{x} \sqrt{x} d x$

Step 1.3.1.6.3

Raise $\sqrt{x}$ to the power of $1$ .

$\int a - \sqrt{x a} - \sqrt{a x} + {\sqrt{x}}^{1} \sqrt{x} d x$

Step 1.3.1.6.4

Raise $\sqrt{x}$ to the power of $1$ .

$\int a - \sqrt{x a} - \sqrt{a x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1} d x$

Step 1.3.1.6.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int a - \sqrt{x a} - \sqrt{a x} + {\sqrt{x}}^{1 + 1} d x$

Step 1.3.1.6.6

Add $1$ and $1$ .

$\int a - \sqrt{x a} - \sqrt{a x} + {\sqrt{x}}^{2} d x$

Step 1.3.1.7

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.3.1.7.1

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

$\int a - \sqrt{x a} - \sqrt{a x} + {(x^{\frac{1}{2}})}^{2} d x$

Step 1.3.1.7.2

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

$\int a - \sqrt{x a} - \sqrt{a x} + x^{\frac{1}{2} \cdot 2} d x$

Step 1.3.1.7.3

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

$\int a - \sqrt{x a} - \sqrt{a x} + x^{\frac{2}{2}} d x$

Step 1.3.1.7.4

Cancel the common factor of $2$ .

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Step 1.3.1.7.4.1

Cancel the common factor.

$\int a - \sqrt{x a} - \sqrt{a x} + x^{\frac{2}{2}} d x$

Step 1.3.1.7.4.2

Rewrite the expression.

$\int a - \sqrt{x a} - \sqrt{a x} + x^{1} d x$

Step 1.3.1.7.5

Simplify.

$\int a - \sqrt{x a} - \sqrt{a x} + x d x$

Step 1.3.2

Subtract $\sqrt{a x}$ from $- \sqrt{x a}$ .

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Step 1.3.2.1

Reorder $x$ and $a$ .

$\int a - \sqrt{a x} - \sqrt{a x} + x d x$

Step 1.3.2.2

Subtract $\sqrt{a x}$ from $- \sqrt{a x}$ .

$\int a - 2 \sqrt{a x} + x d x$

Step 2

Split the single integral into multiple integrals.

$\int a d x + \int - 2 \sqrt{a x} d x + \int x d x$

Step 3

Apply the constant rule.

$a x + C + \int - 2 \sqrt{a x} d x + \int x d x$

Step 4

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$a x + C - 2 \int \sqrt{a x} d x + \int x d x$

Step 5

Let $u = a x$ . Then $d u = a d x$ , so $\frac{1}{a} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = a x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $a x$ .

$\frac{d}{d x} [a x]$

Step 5.1.2

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

$a \frac{d}{d x} [x]$

Step 5.1.3

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

$a \cdot 1$

Step 5.1.4

Multiply $a$ by $1$ .

$a$

Step 5.2

Rewrite the problem using $u$ and $d u$ .

$a x + C - 2 \int \sqrt{u} \frac{1}{a} d u + \int x d x$

Step 6

Combine $\sqrt{u}$ and $\frac{1}{a}$ .

$a x + C - 2 \int \frac{\sqrt{u}}{a} d u + \int x d x$

Step 7

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

$a x + C - 2 (\frac{1}{a} \int \sqrt{u} d u) + \int x d x$

Step 8

Simplify the expression.

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Step 8.1

Simplify.

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Step 8.1.1

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

$a x + C + \frac{- 2}{a} \int \sqrt{u} d u + \int x d x$

Step 8.1.2

Move the negative in front of the fraction.

$a x + C - \frac{2}{a} \int \sqrt{u} d u + \int x d x$

Step 8.2

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

$a x + C - \frac{2}{a} \int u^{\frac{1}{2}} d u + \int x d x$

Step 9

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

$a x + C - \frac{2}{a} (\frac{2}{3} u^{\frac{3}{2}} + C) + \int x d x$

Step 10

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

$a x + C - \frac{2}{a} (\frac{2}{3} u^{\frac{3}{2}} + C) + \frac{1}{2} x^{2} + C$

Step 11

Simplify.

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Step 11.1

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

$a x + C - \frac{2}{a} (\frac{2 u^{\frac{3}{2}}}{3} + C) + \frac{1}{2} x^{2} + C$

Step 11.2

Simplify.

$a x - \frac{4 u^{\frac{3}{2}}}{3 a} + \frac{1}{2} x^{2} + C$

Step 12

Replace all occurrences of $u$ with $a x$ .

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