Calculus Examples

$\int_{0}^{\frac{π}{2}} 2 \tan (x) d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{\frac{π}{2}} \tan (x) d x$

Step 2

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

$2 (\ln (| \sec (x) |)]_{0}^{\frac{π}{2}})$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\ln (| \sec (x) |)$ at $\frac{π}{2}$ and at $0$ .

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

Step 3.2

Simplify.

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Step 3.2.1

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

$2 (\ln (| \sec (\frac{π}{2}) |) - \ln (| 1 |))$

Step 3.2.2

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

$2 \ln (\frac{| \sec (\frac{π}{2}) |}{| 1 |})$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Rewrite $\sec (\frac{π}{2})$ in terms of sines and cosines.

$2 \ln (\frac{| \frac{1}{\cos (\frac{π}{2})} |}{| 1 |})$

Step 3.3.1.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$2 \ln (\frac{| \frac{1}{0} |}{| 1 |})$

Step 3.3.1.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 \ln (\frac{| \frac{1}{0} |}{| 1 |})$

Step 3.3.2

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

$2 \ln (\frac{| \frac{1}{0} |}{1})$

Step 3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 \ln (\frac{| \frac{1}{0} |}{1})$

Step 3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$2 \ln (\frac{| \frac{1}{0} |}{1})$

Step 4

The expression contains a division by $0$ . The expression is undefined.

Undefined