Calculus Examples

$\int_{0}^{5} \frac{1}{\sqrt{9 + 4 x^{2}}} d x$

Step 1

Let $x = \frac{3}{2} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{3 \sec^{2} (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{3 \sec^{2} (t)}{2}$ is positive.

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 + 4 {(\frac{3}{2} \tan (t))}^{2}}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{9 + 4 {(\frac{3}{2} \tan (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $\frac{3}{2}$ and $\tan (t)$ .

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 + 4 {(\frac{3 \tan (t)}{2})}^{2}}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1.2.1

Apply the product rule to $\frac{3 \tan (t)}{2}$ .

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 + 4 \frac{{(3 \tan (t))}^{2}}{2^{2}}}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.2.2

Apply the product rule to $3 \tan (t)$ .

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 + 4 \frac{3^{2} \tan^{2} (t)}{2^{2}}}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.3

Raise $3$ to the power of $2$ .

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 + 4 \frac{9 \tan^{2} (t)}{2^{2}}}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.4

Raise $2$ to the power of $2$ .

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 + 4 \frac{9 \tan^{2} (t)}{4}}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.5

Cancel the common factor of $4$ .

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Step 2.1.1.5.1

Cancel the common factor.

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 + 4 \frac{9 \tan^{2} (t)}{4}}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.5.2

Rewrite the expression.

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 + 9 \tan^{2} (t)}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.2

Factor $9$ out of $9$ .

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 (1) + 9 \tan^{2} (t)}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.3

Factor $9$ out of $9 \tan^{2} (t)$ .

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 (1) + 9 (\tan^{2} (t))}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.4

Factor $9$ out of $9 (1) + 9 (\tan^{2} (t))$ .

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 (1 + \tan^{2} (t))}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.5

Rearrange terms.

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 (\tan^{2} (t) + 1)}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.6

Apply pythagorean identity.

$\int_{0}^{1.27933953} \frac{1}{\sqrt{9 \sec^{2} (t)}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.7

Rewrite $9 \sec^{2} (t)$ as ${(3 \sec (t))}^{2}$ .

$\int_{0}^{1.27933953} \frac{1}{\sqrt{{(3 \sec (t))}^{2}}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.8

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

$\int_{0}^{1.27933953} \frac{1}{3 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $\frac{1}{3 \sec (t)}$ by $\frac{3 \sec^{2} (t)}{2}$ .

$\int_{0}^{1.27933953} \frac{3 \sec^{2} (t)}{3 \sec (t) \cdot 2} d t$

Step 2.2.2

Multiply $2$ by $3$ .

$\int_{0}^{1.27933953} \frac{3 \sec^{2} (t)}{6 \sec (t)} d t$

Step 2.2.3

Cancel the common factor of $3$ and $6$ .

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Step 2.2.3.1

Factor $3$ out of $3 \sec^{2} (t)$ .

$\int_{0}^{1.27933953} \frac{3 (\sec^{2} (t))}{6 \sec (t)} d t$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $3$ out of $6 \sec (t)$ .

$\int_{0}^{1.27933953} \frac{3 (\sec^{2} (t))}{3 (2 \sec (t))} d t$

Step 2.2.3.2.2

Cancel the common factor.

$\int_{0}^{1.27933953} \frac{3 \sec^{2} (t)}{3 (2 \sec (t))} d t$

Step 2.2.3.2.3

Rewrite the expression.

$\int_{0}^{1.27933953} \frac{\sec^{2} (t)}{2 \sec (t)} d t$

Step 2.2.4

Cancel the common factor of $\sec^{2} (t)$ and $\sec (t)$ .

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Step 2.2.4.1

Factor $\sec (t)$ out of $\sec^{2} (t)$ .

$\int_{0}^{1.27933953} \frac{\sec (t) \sec (t)}{2 \sec (t)} d t$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $\sec (t)$ out of $2 \sec (t)$ .

$\int_{0}^{1.27933953} \frac{\sec (t) \sec (t)}{\sec (t) \cdot 2} d t$

Step 2.2.4.2.2

Cancel the common factor.

$\int_{0}^{1.27933953} \frac{\sec (t) \sec (t)}{\sec (t) \cdot 2} d t$

Step 2.2.4.2.3

Rewrite the expression.

$\int_{0}^{1.27933953} \frac{\sec (t)}{2} d t$

Step 3

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{0}^{1.27933953} \sec (t) d t$

Step 4

The integral of $\sec (t)$ with respect to $t$ is $\ln (| \sec (t) + \tan (t) |)$ .

$\frac{1}{2} \ln (| \sec (t) + \tan (t) |)]_{0}^{1.27933953}$

Step 5

Evaluate $\ln (| \sec (t) + \tan (t) |)$ at $1.27933953$ and at $0$ .

$\frac{1}{2} (\ln (| \sec (1.27933953) + \tan (1.27933953) |) - \ln (| \sec (0) + \tan (0) |))$

Step 6

Simplify.

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Step 6.1

The exact value of $\sec (0)$ is $1$ .

$\frac{1}{2} (\ln (| \sec (1.27933953) + \tan (1.27933953) |) - \ln (| 1 + \tan (0) |))$

Step 6.2

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

$\frac{1}{2} (\ln (| \sec (1.27933953) + \tan (1.27933953) |) - \ln (| 1 + 0 |))$

Step 6.3

Add $1$ and $0$ .

$\frac{1}{2} (\ln (| \sec (1.27933953) + \tan (1.27933953) |) - \ln (| 1 |))$

Step 6.4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{1}{2} \ln (\frac{| \sec (1.27933953) + \tan (1.27933953) |}{| 1 |})$

Step 6.5

Combine $\frac{1}{2}$ and $\ln (\frac{| \sec (1.27933953) + \tan (1.27933953) |}{| 1 |})$ .

$\frac{\ln (\frac{| \sec (1.27933953) + \tan (1.27933953) |}{| 1 |})}{2}$

Step 7

Simplify.

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Step 7.1

$\sec (1.27933953) + \tan (1.27933953)$ is approximately $6.8134355$ which is positive so remove the absolute value

$\frac{\ln (\frac{\sec (1.27933953) + \tan (1.27933953)}{| 1 |})}{2}$

Step 7.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{\ln (\frac{\sec (1.27933953) + \tan (1.27933953)}{1})}{2}$

Step 7.3

Divide $\sec (1.27933953) + \tan (1.27933953)$ by $1$ .

$\frac{\ln (\sec (1.27933953) + \tan (1.27933953))}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\sec (1.27933953) + \tan (1.27933953))}{2}$

Decimal Form:

$0.95944823 \dots$

Step 9