Calculus Examples

$\int_{0}^{\sqrt{\frac{3}{2}}} \frac{1}{\sqrt{1 - r^{2}}} d r$

Step 1

Rewrite $1$ as $1^{2}$ .

$\int_{0}^{\sqrt{\frac{3}{2}}} \frac{1}{\sqrt{1^{2} - r^{2}}} d r$

Step 2

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

$\arcsin (r)]_{0}^{\sqrt{\frac{3}{2}}}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\arcsin (r)$ at $\sqrt{\frac{3}{2}}$ and at $0$ .

$\arcsin (\sqrt{\frac{3}{2}}) - \arcsin (0)$

Step 3.2

Simplify.

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Step 3.2.1

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$\arcsin (\frac{\sqrt{3}}{\sqrt{2}}) - \arcsin (0)$

Step 3.2.2

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\arcsin (\frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) - \arcsin (0)$

Step 3.2.3

Combine and simplify the denominator.

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Step 3.2.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}) - \arcsin (0)$

Step 3.2.3.2

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

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}) - \arcsin (0)$

Step 3.2.3.3

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

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}) - \arcsin (0)$

Step 3.2.3.4

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

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}) - \arcsin (0)$

Step 3.2.3.5

Add $1$ and $1$ .

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}) - \arcsin (0)$

Step 3.2.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.2.3.6.1

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

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}) - \arcsin (0)$

Step 3.2.3.6.2

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

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}) - \arcsin (0)$

Step 3.2.3.6.3

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

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}) - \arcsin (0)$

Step 3.2.3.6.4

Cancel the common factor of $2$ .

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Step 3.2.3.6.4.1

Cancel the common factor.

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}) - \arcsin (0)$

Step 3.2.3.6.4.2

Rewrite the expression.

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2^{1}}) - \arcsin (0)$

Step 3.2.3.6.5

Evaluate the exponent.

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2}) - \arcsin (0)$

Step 3.2.4

Combine using the product rule for radicals.

$\arcsin (\frac{\sqrt{3 \cdot 2}}{2}) - \arcsin (0)$

Step 3.2.5

Multiply $3$ by $2$ .

$\arcsin (\frac{\sqrt{6}}{2}) - \arcsin (0)$

Step 3.3

Simplify.

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Step 3.3.1

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

$\arcsin (\frac{\sqrt{6}}{2}) - 0$

Step 3.3.2

Multiply $- 1$ by $0$ .

$\arcsin (\frac{\sqrt{6}}{2}) + 0$

Step 3.3.3

Add $\arcsin (\frac{\sqrt{6}}{2})$ and $0$ .

$\arcsin (\frac{\sqrt{6}}{2})$

Step 4