Calculus Examples

$\int_{0}^{\frac{π}{3}} \sqrt{1 + \tan^{2} (x)} d x$

Step 1

Simplify.

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Step 1.1

Rearrange terms.

$\int_{0}^{\frac{π}{3}} \sqrt{\tan^{2} (x) + 1} d x$

Step 1.2

Apply pythagorean identity.

$\int_{0}^{\frac{π}{3}} \sqrt{\sec^{2} (x)} d x$

Step 1.3

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

$\int_{0}^{\frac{π}{3}} \sec (x) d x$

Step 2

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

$\ln (| \sec (x) + \tan (x) |)]_{0}^{\frac{π}{3}}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\ln (| \sec (x) + \tan (x) |)$ at $\frac{π}{3}$ and at $0$ .

$\ln (| \sec (\frac{π}{3}) + \tan (\frac{π}{3}) |) - \ln (| \sec (0) + \tan (0) |)$

Step 3.2

Simplify.

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Step 3.2.1

The exact value of $\sec (\frac{π}{3})$ is $2$ .

$\ln (| 2 + \tan (\frac{π}{3}) |) - \ln (| \sec (0) + \tan (0) |)$

Step 3.2.2

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$\ln (| 2 + \sqrt{3} |) - \ln (| \sec (0) + \tan (0) |)$

Step 3.2.3

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

$\ln (| 2 + \sqrt{3} |) - \ln (| 1 + \tan (0) |)$

Step 3.2.4

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

$\ln (| 2 + \sqrt{3} |) - \ln (| 1 + 0 |)$

Step 3.2.5

Add $1$ and $0$ .

$\ln (| 2 + \sqrt{3} |) - \ln (| 1 |)$

Step 3.2.6

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

$\ln (\frac{| 2 + \sqrt{3} |}{| 1 |})$

Step 3.3

Simplify.

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Step 3.3.1

$2 + \sqrt{3}$ is approximately $3.7320508$ which is positive so remove the absolute value

$\ln (\frac{2 + \sqrt{3}}{| 1 |})$

Step 3.3.2

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

$\ln (\frac{2 + \sqrt{3}}{1})$

Step 3.3.3

Divide $2 + \sqrt{3}$ by $1$ .

$\ln (2 + \sqrt{3})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\ln (2 + \sqrt{3})$

Decimal Form:

$1.31695789 \dots$