Calculus Examples

$\sin (\arctan (x))$

Step 1

Draw a triangle in the plane with vertices $(1, x)$ , $(1, 0)$ , and the origin. Then $\arctan (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, x)$ . Therefore, $\sin (\arctan (x))$ is $\frac{x}{\sqrt{1 + x^{2}}}$ .

$\frac{x}{\sqrt{1 + x^{2}}}$

Step 2

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

$\frac{x}{\sqrt{1 + x^{2}}} \cdot \frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

$\frac{x \sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}} \sqrt{1 + x^{2}}}$

Step 3.2

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

$\frac{x \sqrt{1 + x^{2}}}{{\sqrt{1 + x^{2}}}^{1} \sqrt{1 + x^{2}}}$

Step 3.3

Raise $\sqrt{1 + x^{2}}$ to the power of $1$ .

$\frac{x \sqrt{1 + x^{2}}}{{\sqrt{1 + x^{2}}}^{1} {\sqrt{1 + x^{2}}}^{1}}$

Step 3.4

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

$\frac{x \sqrt{1 + x^{2}}}{{\sqrt{1 + x^{2}}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{x \sqrt{1 + x^{2}}}{{\sqrt{1 + x^{2}}}^{2}}$

Step 3.6

Rewrite ${\sqrt{1 + x^{2}}}^{2}$ as $1 + x^{2}$ .

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Step 3.6.1

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

$\frac{x \sqrt{1 + x^{2}}}{{({(1 + x^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{x \sqrt{1 + x^{2}}}{{(1 + x^{2})}^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{x \sqrt{1 + x^{2}}}{{(1 + x^{2})}^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{x \sqrt{1 + x^{2}}}{{(1 + x^{2})}^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{x \sqrt{1 + x^{2}}}{{(1 + x^{2})}^{1}}$

Step 3.6.5

Simplify.

$\frac{x \sqrt{1 + x^{2}}}{1 + x^{2}}$