Calculus Examples

$\int_{0}^{0.5} \frac{3}{\sqrt{1 - x^{2}}} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{0}^{0.5} \frac{1}{\sqrt{1 - x^{2}}} d x$

Step 2

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

$3 \int_{0}^{0.5} \frac{1}{\sqrt{1^{2} - x^{2}}} d x$

Step 3

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

$3 (\arcsin (x)]_{0}^{0.5})$

Step 4

Evaluate $\arcsin (x)$ at $0.5$ and at $0$ .

$3 (\arcsin (0.5) - \arcsin (0))$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Evaluate $\arcsin (0.5)$ .

$3 (\frac{π}{6} - \arcsin (0))$

Step 5.1.2

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

$3 (\frac{π}{6} - 0)$

Step 5.1.3

Multiply $- 1$ by $0$ .

$3 (\frac{π}{6} + 0)$

Step 5.2

Add $\frac{π}{6}$ and $0$ .

$3 \frac{π}{6}$

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Factor $3$ out of $6$ .

$3 \frac{π}{3 (2)}$

Step 5.3.2

Cancel the common factor.

$3 \frac{π}{3 \cdot 2}$

Step 5.3.3

Rewrite the expression.

$\frac{π}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2}$

Decimal Form:

$1.57079632 \dots$

Step 7