Calculus Examples

$a^{2} d x = x \sqrt{x^{2} a^{2}} d y$

Step 1

Rewrite the equation.

$x \sqrt{x^{2} a^{2}} d y = a^{2} d x$

Step 2

Multiply both sides by $\frac{1}{x \sqrt{x^{2} a^{2}}}$ .

$\frac{1}{x \sqrt{x^{2} a^{2}}} (x \sqrt{x^{2} a^{2}}) d y = \frac{1}{x \sqrt{x^{2} a^{2}}} a^{2} d x$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x \sqrt{x^{2} a^{2}}$ .

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Step 3.1.1

Cancel the common factor.

$\frac{1}{x \sqrt{x^{2} a^{2}}} (x \sqrt{x^{2} a^{2}}) d y = \frac{1}{x \sqrt{x^{2} a^{2}}} a^{2} d x$

Step 3.1.2

Rewrite the expression.

$1 d y = \frac{1}{x \sqrt{x^{2} a^{2}}} a^{2} d x$

Step 3.2

Simplify the denominator.

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Step 3.2.1

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

$1 d y = \frac{1}{x \sqrt{{(x a)}^{2}}} a^{2} d x$

Step 3.2.2

Pull terms out from under the radical, assuming positive real numbers.

$1 d y = \frac{1}{x \cdot x a} a^{2} d x$

Step 3.2.3

Combine exponents.

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Step 3.2.3.1

Raise $x$ to the power of $1$ .

$1 d y = \frac{1}{x^{1} x a} a^{2} d x$

Step 3.2.3.2

Raise $x$ to the power of $1$ .

$1 d y = \frac{1}{x^{1} x^{1} a} a^{2} d x$

Step 3.2.3.3

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

$1 d y = \frac{1}{x^{1 + 1} a} a^{2} d x$

Step 3.2.3.4

Add $1$ and $1$ .

$1 d y = \frac{1}{x^{2} a} a^{2} d x$

Step 3.3

Cancel the common factor of $a$ .

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Step 3.3.1

Factor $a$ out of $x^{2} a$ .

$1 d y = \frac{1}{a x^{2}} a^{2} d x$

Step 3.3.2

Factor $a$ out of $a^{2}$ .

$1 d y = \frac{1}{a x^{2}} (a \cdot a) d x$

Step 3.3.3

Cancel the common factor.

$1 d y = \frac{1}{a x^{2}} (a \cdot a) d x$

Step 3.3.4

Rewrite the expression.

$1 d y = \frac{1}{x^{2}} a d x$

Step 3.4

Combine $\frac{1}{x^{2}}$ and $a$ .

$1 d y = \frac{a}{x^{2}} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int d y = \int \frac{a}{x^{2}} d x$

Step 4.2

Apply the constant rule.

$y + C_{1} = \int \frac{a}{x^{2}} d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

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

$y + C_{1} = a \int \frac{1}{x^{2}} d x$

Step 4.3.2

Apply basic rules of exponents.

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Step 4.3.2.1

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

$y + C_{1} = a \int {(x^{2})}^{- 1} d x$

Step 4.3.2.2

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

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Step 4.3.2.2.1

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

$y + C_{1} = a \int x^{2 \cdot - 1} d x$

Step 4.3.2.2.2

Multiply $2$ by $- 1$ .

$y + C_{1} = a \int x^{- 2} d x$

Step 4.3.3

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

$y + C_{1} = a (- x^{- 1} + C_{2})$

Step 4.3.4

Simplify the answer.

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Step 4.3.4.1

Rewrite $a (- x^{- 1} + C_{2})$ as $a (- \frac{1}{x}) + C_{2}$ .

$y + C_{1} = a (- \frac{1}{x}) + C_{2}$

Step 4.3.4.2

Combine $a$ and $\frac{1}{x}$ .

$y + C_{1} = - \frac{a}{x} + C_{2}$

Step 4.4

Group the constant of integration on the right side as $K$ .

$y = - \frac{a}{x} + K$