Calculus Examples

$\int_{0}^{\frac{π}{6}} \sec (2 x) \tan (2 x) d x$

Step 1

Let $u_{2} = \sec (2 x)$ . Then $d u_{2} = 2 \sec (2 x) \tan (2 x) d x$ , so $\frac{1}{2} d u_{2} = \sec (2 x) \tan (2 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \sec (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\sec (2 x)$ .

$\frac{d}{d x} [\sec (2 x)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 2 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d x} [2 x]$

Step 1.1.2.2

The derivative of $\sec (u_{1})$ with respect to $u_{1}$ is $\sec (u_{1}) \tan (u_{1})$ .

$\sec (u_{1}) \tan (u_{1}) \frac{d}{d x} [2 x]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$\sec (2 x) \tan (2 x) \frac{d}{d x} [2 x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$\sec (2 x) \tan (2 x) (2 \frac{d}{d x} [x])$

Step 1.1.3.2

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

$\sec (2 x) \tan (2 x) (2 \cdot 1)$

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply $2$ by $1$ .

$\sec (2 x) \tan (2 x) \cdot 2$

Step 1.1.3.3.2

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

$2 \sec (2 x) \tan (2 x)$

Step 1.2

Substitute the lower limit in for $x$ in $u_{2} = \sec (2 x)$ .

$u_{lower} = \sec (2 \cdot 0)$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $2$ by $0$ .

$u_{lower} = \sec (0)$

Step 1.3.2

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

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u_{2} = \sec (2 x)$ .

$u_{upper} = \sec (2 \frac{π}{6})$

Step 1.5

Simplify.

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Step 1.5.1

Cancel the common factor of $2$ .

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Step 1.5.1.1

Factor $2$ out of $6$ .

$u_{upper} = \sec (2 \frac{π}{2 (3)})$

Step 1.5.1.2

Cancel the common factor.

$u_{upper} = \sec (2 \frac{π}{2 \cdot 3})$

Step 1.5.1.3

Rewrite the expression.

$u_{upper} = \sec (\frac{π}{3})$

Step 1.5.2

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

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 1.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\int_{1}^{2} \frac{1}{2} d u_{2}$

Step 2

Apply the constant rule.

$\frac{1}{2} u_{2}]_{1}^{2}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{2} u_{2}$ at $2$ and at $1$ .

$(\frac{1}{2} \cdot 2) - \frac{1}{2} \cdot 1$

Step 3.2

Simplify.

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Step 3.2.1

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

$\frac{2}{2} - \frac{1}{2} \cdot 1$

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{2}{2} - \frac{1}{2} \cdot 1$

Step 3.2.2.2

Rewrite the expression.

$1 - \frac{1}{2} \cdot 1$

Step 3.2.3

Multiply $- 1$ by $1$ .

$1 - \frac{1}{2}$

Step 3.2.4

Write $1$ as a fraction with a common denominator.

$\frac{2}{2} - \frac{1}{2}$

Step 3.2.5

Combine the numerators over the common denominator.

$\frac{2 - 1}{2}$

Step 3.2.6

Subtract $1$ from $2$ .

$\frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$