Calculus Examples

sin (arctan (x))

$\sin (\arctan (x))$

Step 1

Draw a triangle in the plane with vertices

(1, x)

$(1, x)$ ,

(1, 0)

$(1, 0)$ , and the origin. Then

arctan (x)

$\arctan (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

(1, x)

$(1, x)$ . Therefore,

sin (arctan (x))

$\sin (\arctan (x))$ is

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

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

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

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

Step 2

Multiply

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

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

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

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

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply

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

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

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

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

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

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

1

$1$ .

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

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

Step 3.3

Raise

\sqrt{1 + x^{2}}

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

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.5

Add

1

$1$ and

1

$1$ .

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

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

Step 3.6

Rewrite

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

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

1 + x^{2}

$1 + x^{2}$ .

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Step 3.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{1 + x^{2}}

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

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

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

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 3.6.4

Cancel the common factor of

2

$2$ .

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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.

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