Calculus Examples

$\int_{- \frac{1}{2}}^{0} \frac{14}{\sqrt{25 - 25 x^{2}}} d x$

Step 1

Since $14$ is constant with respect to $x$ , move $14$ out of the integral.

$14 \int_{- \frac{1}{2}}^{0} \frac{1}{\sqrt{25 - 25 x^{2}}} d x$

Step 2

Let $x = \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\cos (t)$ is positive.

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{\sqrt{25 - 25 \sin^{2} (t)}} \cos (t) d t$

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{25 - 25 \sin^{2} (t)}$ .

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Step 3.1.1

Factor $25$ out of $25$ .

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{\sqrt{25 (1) - 25 \sin^{2} (t)}} \cos (t) d t$

Step 3.1.2

Factor $25$ out of $- 25 \sin^{2} (t)$ .

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{\sqrt{25 (1) + 25 (- \sin^{2} (t))}} \cos (t) d t$

Step 3.1.3

Factor $25$ out of $25 (1) + 25 (- \sin^{2} (t))$ .

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{\sqrt{25 (1 - \sin^{2} (t))}} \cos (t) d t$

Step 3.1.4

Apply pythagorean identity.

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{\sqrt{25 \cos^{2} (t)}} \cos (t) d t$

Step 3.1.5

Rewrite $25 \cos^{2} (t)$ as ${(5 \cos (t))}^{2}$ .

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{\sqrt{{(5 \cos (t))}^{2}}} \cos (t) d t$

Step 3.1.6

Pull terms out from under the radical, assuming positive real numbers.

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{5 \cos (t)} \cos (t) d t$

Step 3.2

Cancel the common factor of $\cos (t)$ .

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Step 3.2.1

Factor $\cos (t)$ out of $5 \cos (t)$ .

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{\cos (t) \cdot 5} \cos (t) d t$

Step 3.2.2

Cancel the common factor.

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{\cos (t) \cdot 5} \cos (t) d t$

Step 3.2.3

Rewrite the expression.

$14 \int_{- \frac{π}{6}}^{0} \frac{1}{5} d t$

Step 4

Apply the constant rule.

$14 (\frac{1}{5} t]_{- \frac{π}{6}}^{0})$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{5}$ and $t$ .

$14 (\frac{t}{5}]_{- \frac{π}{6}}^{0})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{t}{5}$ at $0$ and at $- \frac{π}{6}$ .

$14 ((\frac{0}{5}) - \frac{- \frac{π}{6}}{5})$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Cancel the common factor of $0$ and $5$ .

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Step 5.2.2.1.1

Factor $5$ out of $0$ .

$14 (\frac{5 (0)}{5} - \frac{- \frac{π}{6}}{5})$

Step 5.2.2.1.2

Cancel the common factors.

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Step 5.2.2.1.2.1

Factor $5$ out of $5$ .

$14 (\frac{5 \cdot 0}{5 \cdot 1} - \frac{- \frac{π}{6}}{5})$

Step 5.2.2.1.2.2

Cancel the common factor.

$14 (\frac{5 \cdot 0}{5 \cdot 1} - \frac{- \frac{π}{6}}{5})$

Step 5.2.2.1.2.3

Rewrite the expression.

$14 (\frac{0}{1} - \frac{- \frac{π}{6}}{5})$

Step 5.2.2.1.2.4

Divide $0$ by $1$ .

$14 (0 - \frac{- \frac{π}{6}}{5})$

Step 5.2.2.2

Rewrite $\frac{- \frac{π}{6}}{5}$ as a product.

$14 (0 - (- \frac{π}{6} \cdot \frac{1}{5}))$

Step 5.2.2.3

Multiply $\frac{1}{5}$ by $\frac{π}{6}$ .

$14 (0 - - \frac{π}{5 \cdot 6})$

Step 5.2.2.4

Multiply $5$ by $6$ .

$14 (0 - - \frac{π}{30})$

Step 5.2.2.5

Multiply $- 1$ by $- 1$ .

$14 (0 + 1 \frac{π}{30})$

Step 5.2.2.6

Multiply $\frac{π}{30}$ by $1$ .

$14 (0 + \frac{π}{30})$

Step 5.2.2.7

Add $0$ and $\frac{π}{30}$ .

$14 \frac{π}{30}$

Step 5.2.2.8

Combine $14$ and $\frac{π}{30}$ .

$\frac{14 π}{30}$

Step 5.2.2.9

Cancel the common factor of $14$ and $30$ .

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Step 5.2.2.9.1

Factor $2$ out of $14 π$ .

$\frac{2 (7 π)}{30}$

Step 5.2.2.9.2

Cancel the common factors.

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Step 5.2.2.9.2.1

Factor $2$ out of $30$ .

$\frac{2 (7 π)}{2 (15)}$

Step 5.2.2.9.2.2

Cancel the common factor.

$\frac{2 (7 π)}{2 \cdot 15}$

Step 5.2.2.9.2.3

Rewrite the expression.

$\frac{7 π}{15}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{7 π}{15}$

Decimal Form:

$1.46607657 \dots$

Step 7