Calculus Examples

$\int_{0}^{0.5} 3 \tan (2 x) d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{0}^{0.5} \tan (2 x) d x$

Step 2

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.1.3

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

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 2.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \cdot 0.5$

Step 2.5

Multiply $2$ by $0.5$ .

$u_{upper} = 1$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 2.7

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

$3 \int_{0}^{1} \tan (u) \frac{1}{2} d u$

Step 3

Combine $\tan (u)$ and $\frac{1}{2}$ .

$3 \int_{0}^{1} \frac{\tan (u)}{2} d u$

Step 4

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

$3 (\frac{1}{2} \int_{0}^{1} \tan (u) d u)$

Step 5

Combine $\frac{1}{2}$ and $3$ .

$\frac{3}{2} \int_{0}^{1} \tan (u) d u$

Step 6

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

$\frac{3}{2} \ln (| \sec (u) |)]_{0}^{1}$

Step 7

Evaluate $\ln (| \sec (u) |)$ at $1$ and at $0$ .

$\frac{3}{2} (\ln (| \sec (1) |) - \ln (| \sec (0) |))$

Step 8

Simplify.

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Step 8.1

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

$\frac{3}{2} (\ln (| \sec (1) |) - \ln (| 1 |))$

Step 8.2

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

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

Step 8.3

Combine $\frac{3}{2}$ and $\ln (\frac{| \sec (1) |}{| 1 |})$ .

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.00022847 \dots$