Calculus Examples

$\int_{1}^{8 \sqrt{3}} \frac{11}{\sqrt{256 - x^{2}}} d x$

Step 1

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$11 \int_{1}^{8 \sqrt{3}} \frac{1}{\sqrt{256 - x^{2}}} d x$

Step 2

Rewrite $256$ as $16^{2}$ .

$11 \int_{1}^{8 \sqrt{3}} \frac{1}{\sqrt{16^{2} - x^{2}}} d x$

Step 3

The integral of $\frac{1}{\sqrt{16^{2} - x^{2}}}$ with respect to $x$ is $\arcsin (\frac{x}{16})$

$11 (\arcsin (\frac{x}{16})]_{1}^{8 \sqrt{3}})$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\arcsin (\frac{x}{16})$ at $8 \sqrt{3}$ and at $1$ .

$11 ((\arcsin (\frac{8 \sqrt{3}}{16})) - \arcsin (\frac{1}{16}))$

Step 4.2

Cancel the common factor of $8$ and $16$ .

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Step 4.2.1

Factor $8$ out of $8 \sqrt{3}$ .

$11 (\arcsin (\frac{8 (\sqrt{3})}{16}) - \arcsin (\frac{1}{16}))$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $8$ out of $16$ .

$11 (\arcsin (\frac{8 \sqrt{3}}{8 \cdot 2}) - \arcsin (\frac{1}{16}))$

Step 4.2.2.2

Cancel the common factor.

$11 (\arcsin (\frac{8 \sqrt{3}}{8 \cdot 2}) - \arcsin (\frac{1}{16}))$

Step 4.2.2.3

Rewrite the expression.

$11 (\arcsin (\frac{\sqrt{3}}{2}) - \arcsin (\frac{1}{16}))$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $\frac{π}{3}$ .

$11 (\frac{π}{3} - \arcsin (\frac{1}{16}))$

Step 5.1.2

Evaluate $\arcsin (\frac{1}{16})$ .

$11 (\frac{π}{3} - 1 \cdot 0.06254076)$

Step 5.1.3

Multiply $- 1$ by $0.06254076$ .

$11 (\frac{π}{3} - 0.06254076)$

Step 5.2

Subtract $0.06254076$ from $\frac{π}{3}$ .

$11 \cdot 0.98465678$

Step 5.3

Multiply $11$ by $0.98465678$ .

$10.83122468$

Step 6