Calculus Examples

$\int \frac{\arccos (2 x)}{\sqrt{1 - 4 x^{2}}} d x$

Step 1

Let $u_{2} = \arccos (2 x)$ . Then $d u_{2} = - \frac{2}{\sqrt{1 - 4 x^{2}}} d x$ , so $1 d u_{2} = - \frac{2}{\sqrt{1 - 4 x^{2}}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \arccos (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\arccos (2 x)$ .

$\frac{d}{d x} [\arccos (2 x)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arccos (x)$ and $g (x) = 2 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{d}{d u_{1}} [\arccos (u_{1})] \frac{d}{d x} [2 x]$

Step 1.1.2.2

The derivative of $\arccos (u_{1})$ with respect to $u_{1}$ is $- \frac{1}{\sqrt{1 - {u_{1}}^{2}}}$ .

$- \frac{1}{\sqrt{1 - {u_{1}}^{2}}} \frac{d}{d x} [2 x]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$- \frac{1}{\sqrt{1 - {(2 x)}^{2}}} \frac{d}{d x} [2 x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Factor $2$ out of $2 x$ .

$- \frac{1}{\sqrt{1 - {(2 (x))}^{2}}} \frac{d}{d x} [2 x]$

Step 1.1.3.2

Simplify the expression.

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Step 1.1.3.2.1

Apply the product rule to $2 (x)$ .

$- \frac{1}{\sqrt{1 - (2^{2} x^{2})}} \frac{d}{d x} [2 x]$

Step 1.1.3.2.2

Raise $2$ to the power of $2$ .

$- \frac{1}{\sqrt{1 - (4 x^{2})}} \frac{d}{d x} [2 x]$

Step 1.1.3.2.3

Multiply $4$ by $- 1$ .

$- \frac{1}{\sqrt{1 - 4 x^{2}}} \frac{d}{d x} [2 x]$

Step 1.1.3.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- \frac{1}{\sqrt{1 - 4 x^{2}}} (2 \frac{d}{d x} [x])$

Step 1.1.3.4

Combine fractions.

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Step 1.1.3.4.1

Multiply $2$ by $- 1$ .

$- 2 \frac{1}{\sqrt{1 - 4 x^{2}}} \frac{d}{d x} [x]$

Step 1.1.3.4.2

Combine $- 2$ and $\frac{1}{\sqrt{1 - 4 x^{2}}}$ .

$\frac{- 2}{\sqrt{1 - 4 x^{2}}} \frac{d}{d x} [x]$

Step 1.1.3.4.3

Move the negative in front of the fraction.

$- \frac{2}{\sqrt{1 - 4 x^{2}}} \frac{d}{d x} [x]$

Step 1.1.3.5

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

$- \frac{2}{\sqrt{1 - 4 x^{2}}} \cdot 1$

Step 1.1.3.6

Multiply $- 1$ by $1$ .

$- \frac{2}{\sqrt{1 - 4 x^{2}}}$

Step 1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int - \frac{1}{2} u_{2} d u_{2}$

Step 2

Since $- \frac{1}{2}$ is constant with respect to $u_{2}$ , move $- \frac{1}{2}$ out of the integral.

$- \frac{1}{2} \int u_{2} d u_{2}$

Step 3

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

$- \frac{1}{2} (\frac{1}{2} {u_{2}}^{2} + C)$

Step 4

Simplify.

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Step 4.1

Rewrite $- \frac{1}{2} (\frac{1}{2} {u_{2}}^{2} + C)$ as $- \frac{1}{2} \cdot \frac{1}{2} {u_{2}}^{2} + C$ .

$- \frac{1}{2} \cdot \frac{1}{2} {u_{2}}^{2} + C$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$- \frac{1}{2 \cdot 2} {u_{2}}^{2} + C$

Step 4.2.2

Multiply $2$ by $2$ .

$- \frac{1}{4} {u_{2}}^{2} + C$

Step 5

Replace all occurrences of $u_{2}$ with $\arccos (2 x)$ .

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