Calculus Examples

$\int_{0.25}^{0.5} \frac{1}{\sqrt{25 - 16 x^{2}}} d x$

Step 1

Write the expression using exponents.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\int_{0.25}^{0.5} \frac{1}{\sqrt{5^{2} - 16 x^{2}}} d x$

Step 1.2

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$\int_{0.25}^{0.5} \frac{1}{\sqrt{5^{2} - {(4 x)}^{2}}} d x$

Step 2

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 2.1.2

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

$4 \frac{d}{d x} [x]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$4 \cdot 1$

Step 2.1.4

Multiply $4$ by $1$ .

$4$

Step 2.2

Substitute the lower limit in for $x$ in $u = 4 x$ .

$u_{lower} = 4 \cdot 0.25$

Step 2.3

Multiply $4$ by $0.25$ .

$u_{lower} = 1$

Step 2.4

Substitute the upper limit in for $x$ in $u = 4 x$ .

$u_{upper} = 4 \cdot 0.5$

Step 2.5

Multiply $4$ by $0.5$ .

$u_{upper} = 2$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{2} \frac{1}{\sqrt{5^{2} - u^{2}}} \cdot \frac{1}{4} d u$

Step 3

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

$\frac{1}{4} \int_{1}^{2} \frac{1}{\sqrt{5^{2} - u^{2}}} d u$

Step 4

The integral of $\frac{1}{\sqrt{5^{2} - u^{2}}}$ with respect to $u$ is $\arcsin (\frac{u}{5})$

$\frac{1}{4} \arcsin (\frac{u}{5})]_{1}^{2}$

Step 5

Combine $\frac{1}{4}$ and $\arcsin (\frac{u}{5})]_{1}^{2}$ .

$\frac{\arcsin (\frac{u}{5})]_{1}^{2}}{4}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\arcsin (\frac{u}{5})$ at $2$ and at $1$ .

$\frac{(\arcsin (\frac{2}{5})) - \arcsin (\frac{1}{5})}{4}$

Step 6.2

Cancel the common factor of $2$ and $5$ .

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Step 6.2.1

Rewrite $2$ as $1 (2)$ .

$\frac{\arcsin (\frac{1 (2)}{5}) - \arcsin (\frac{1}{5})}{4}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Rewrite $5$ as $1 (5)$ .

$\frac{\arcsin (\frac{1 \cdot 2}{1 \cdot 5}) - \arcsin (\frac{1}{5})}{4}$

Step 6.2.2.2

Cancel the common factor.

$\frac{\arcsin (\frac{1 \cdot 2}{1 \cdot 5}) - \arcsin (\frac{1}{5})}{4}$

Step 6.2.2.3

Rewrite the expression.

$\frac{\arcsin (\frac{2}{5}) - \arcsin (\frac{1}{5})}{4}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Evaluate $\arcsin (\frac{2}{5})$ .

$\frac{0.41151684 - \arcsin (\frac{1}{5})}{4}$

Step 7.1.2

Evaluate $\arcsin (\frac{1}{5})$ .

$\frac{0.41151684 - 1 \cdot 0.20135792}{4}$

Step 7.1.3

Multiply $- 1$ by $0.20135792$ .

$\frac{0.41151684 - 0.20135792}{4}$

Step 7.1.4

Subtract $0.20135792$ from $0.41151684$ .

$\frac{0.21015892}{4}$

Step 7.2

Divide $0.21015892$ by $4$ .

$0.05253973$

Step 8