Calculus Examples

$\int_{0}^{\frac{1}{8}} \frac{\arcsin (4 x)}{\sqrt{1 - 16 x^{2}}} d x$

Step 1

Let $u_{2} = \arcsin (4 x)$ . Then $d u_{2} = \frac{4}{\sqrt{1 - 16 x^{2}}} d x$ , so $- d u_{2} = \frac{- 4}{\sqrt{1 - 16 x^{2}}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \arcsin (4 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\arcsin (4 x)$ .

$\frac{d}{d x} [\arcsin (4 x)]$

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 $f (x) = \arcsin (x)$ and $g (x) = 4 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $4 x$ .

$\frac{d}{d u_{1}} [\arcsin (u_{1})] \frac{d}{d x} [4 x]$

Step 1.1.2.2

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

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

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $4 x$ .

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Factor $4$ out of $4 x$ .

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

Step 1.1.3.2

Simplify the expression.

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Step 1.1.3.2.1

Apply the product rule to $4 (x)$ .

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

Step 1.1.3.2.2

Raise $4$ to the power of $2$ .

$\frac{1}{\sqrt{1 - (16 x^{2})}} \frac{d}{d x} [4 x]$

Step 1.1.3.2.3

Multiply $16$ by $- 1$ .

$\frac{1}{\sqrt{1 - 16 x^{2}}} \frac{d}{d x} [4 x]$

Step 1.1.3.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{1}{\sqrt{1 - 16 x^{2}}} (4 \frac{d}{d x} [x])$

Step 1.1.3.4

Combine $4$ and $\frac{1}{\sqrt{1 - 16 x^{2}}}$ .

$\frac{4}{\sqrt{1 - 16 x^{2}}} \frac{d}{d x} [x]$

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 $n = 1$ .

$\frac{4}{\sqrt{1 - 16 x^{2}}} \cdot 1$

Step 1.1.3.6

Multiply $\frac{4}{\sqrt{1 - 16 x^{2}}}$ by $1$ .

$\frac{4}{\sqrt{1 - 16 x^{2}}}$

Step 1.2

Substitute the lower limit in for $x$ in $u_{2} = \arcsin (4 x)$ .

$u_{lower} = \arcsin (4 \cdot 0)$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $4$ by $0$ .

$u_{lower} = \arcsin (0)$

Step 1.3.2

The exact value of $\arcsin (0)$ is $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u_{2} = \arcsin (4 x)$ .

$u_{upper} = \arcsin (4 (\frac{1}{8}))$

Step 1.5

Simplify.

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Step 1.5.1

Cancel the common factor of $4$ .

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Step 1.5.1.1

Factor $4$ out of $8$ .

$u_{upper} = \arcsin (4 \frac{1}{4 (2)})$

Step 1.5.1.2

Cancel the common factor.

$u_{upper} = \arcsin (4 \frac{1}{4 \cdot 2})$

Step 1.5.1.3

Rewrite the expression.

$u_{upper} = \arcsin (\frac{1}{2})$

Step 1.5.2

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$u_{upper} = \frac{π}{6}$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = \frac{π}{6}$

Step 1.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\int_{0}^{\frac{π}{6}} \frac{1}{4} u_{2} d u_{2}$

Step 2

Since $\frac{1}{4}$ is constant with respect to $u_{2}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{0}^{\frac{π}{6}} u_{2} d u_{2}$

Step 3

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

$\frac{1}{4} \frac{1}{2} {u_{2}}^{2}]_{0}^{\frac{π}{6}}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{1}{2} {u_{2}}^{2}$ at $\frac{π}{6}$ and at $0$ .

$\frac{1}{4} ((\frac{1}{2} {(\frac{π}{6})}^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 4.2

Simplify.

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Step 4.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{4} (\frac{1}{2} {(\frac{π}{6})}^{2} - \frac{1}{2} \cdot 0)$

Step 4.2.2

Multiply $0$ by $- 1$ .

$\frac{1}{4} (\frac{1}{2} {(\frac{π}{6})}^{2} + 0 (\frac{1}{2}))$

Step 4.2.3

Multiply $0$ by $\frac{1}{2}$ .

$\frac{1}{4} (\frac{1}{2} {(\frac{π}{6})}^{2} + 0)$

Step 4.2.4

Add $\frac{1}{2} {(\frac{π}{6})}^{2}$ and $0$ .

$\frac{1}{4} (\frac{1}{2} {(\frac{π}{6})}^{2})$

Step 4.2.5

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$\frac{1}{2 \cdot 4} {(\frac{π}{6})}^{2}$

Step 4.2.6

Multiply $2$ by $4$ .

$\frac{1}{8} {(\frac{π}{6})}^{2}$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $\frac{π}{6}$ .

$\frac{1}{8} \cdot \frac{π^{2}}{6^{2}}$

Step 5.2

Combine.

$\frac{1 π^{2}}{8 \cdot 6^{2}}$

Step 5.3

Multiply $π^{2}$ by $1$ .

$\frac{π^{2}}{8 \cdot 6^{2}}$

Step 5.4

Raise $6$ to the power of $2$ .

$\frac{π^{2}}{8 \cdot 36}$

Step 5.5

Multiply $8$ by $36$ .

$\frac{π^{2}}{288}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{π^{2}}{288}$

Decimal Form:

$0.03426945 \dots$