Calculus Examples

$\ln (\sqrt{x^{2} - 3} - x)$

Step 1

Set the argument in $\ln (\sqrt{x^{2} - 3} - x)$ greater than $0$ to find where the expression is defined.

$\sqrt{x^{2} - 3} - x > 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $x$ to both sides of the inequality.

$\sqrt{x^{2} - 3} > x$

Step 2.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{x^{2} - 3}}^{2} > x^{2}$

Step 2.3

Simplify each side of the inequality.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 3}$ as ${(x^{2} - 3)}^{\frac{1}{2}}$ .

${({(x^{2} - 3)}^{\frac{1}{2}})}^{2} > x^{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify ${({(x^{2} - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1

Multiply the exponents in ${({(x^{2} - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.2.1.1.1

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

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

Step 2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

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

Step 2.3.2.1.1.2.2

Rewrite the expression.

${(x^{2} - 3)}^{1} > x^{2}$

Step 2.3.2.1.2

Simplify.

$x^{2} - 3 > x^{2}$

Step 2.4

Solve for $x$ .

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Step 2.4.1

Move all terms containing $x$ to the left side of the inequality.

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Step 2.4.1.1

Subtract $x^{2}$ from both sides of the inequality.

$x^{2} - 3 - x^{2} > 0$

Step 2.4.1.2

Combine the opposite terms in $x^{2} - 3 - x^{2}$ .

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Step 2.4.1.2.1

Subtract $x^{2}$ from $x^{2}$ .

$0 - 3 > 0$

Step 2.4.1.2.2

Subtract $3$ from $0$ .

$- 3 > 0$

Step 2.4.2

Since $- 3 < 0$ , there are no solutions.

No solution

Step 3

Set the radicand in $\sqrt{x^{2} - 3}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} - 3 \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $3$ to both sides of the inequality.

$x^{2} \geq 3$

Step 4.2

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \geq \sqrt{3}$

Step 4.3

Simplify the left side.

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Step 4.3.1

Pull terms out from under the radical.

$| x | \geq \sqrt{3}$

Step 4.4

Write $| x | \geq \sqrt{3}$ as a piecewise.

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Step 4.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 4.4.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \geq \sqrt{3}$

Step 4.4.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 4.4.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \geq \sqrt{3}$

Step 4.4.5

Write as a piecewise.

${\begin{cases} x \geq \sqrt{3} & x \geq 0 \\ - x \geq \sqrt{3} & x < 0 \end{cases}$

Step 4.5

Find the intersection of $x \geq \sqrt{3}$ and $x \geq 0$ .

$x \geq \sqrt{3}$

Step 4.6

Divide each term in $- x \geq \sqrt{3}$ by $- 1$ and simplify.

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Step 4.6.1

Divide each term in $- x \geq \sqrt{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{\sqrt{3}}{- 1}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{\sqrt{3}}{- 1}$

Step 4.6.2.2

Divide $x$ by $1$ .

$x \leq \frac{\sqrt{3}}{- 1}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Move the negative one from the denominator of $\frac{\sqrt{3}}{- 1}$ .

$x \leq - 1 \cdot \sqrt{3}$

Step 4.6.3.2

Rewrite $- 1 \cdot \sqrt{3}$ as $- \sqrt{3}$ .

$x \leq - \sqrt{3}$

Step 4.7

Find the union of the solutions.

$x \leq - \sqrt{3}$ or $x \geq \sqrt{3}$

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - \sqrt{3}] \cup [\sqrt{3}, \infty)$

Set-Builder Notation:

${x | x \leq - \sqrt{3}, x \geq \sqrt{3}}$

Step 6