Calculus Examples

$\int_{0}^{\frac{π}{8}} \frac{\sec^{2} (2 x)}{3 + \tan (2 x)} d x$

Step 1

Let $u_{2} = 3 + \tan (2 x)$ . Then $d u_{2} = 2 \sec^{2} (2 x) d x$ , so $\frac{1}{2} d u_{2} = \sec^{2} (2 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = 3 + \tan (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $3 + \tan (2 x)$ .

$\frac{d}{d x} [3 + \tan (2 x)]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $3 + \tan (2 x)$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [\tan (2 x)]$ .

$\frac{d}{d x} [3] + \frac{d}{d x} [\tan (2 x)]$

Step 1.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\tan (2 x)]$

Step 1.1.3

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

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Step 1.1.3.1

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

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Step 1.1.3.1.1

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

$0 + \frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [2 x]$

Step 1.1.3.1.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

$0 + \sec^{2} (u_{1}) \frac{d}{d x} [2 x]$

Step 1.1.3.1.3

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

$0 + \sec^{2} (2 x) \frac{d}{d x} [2 x]$

Step 1.1.3.2

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

$0 + \sec^{2} (2 x) (2 \frac{d}{d x} [x])$

Step 1.1.3.3

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

$0 + \sec^{2} (2 x) (2 \cdot 1)$

Step 1.1.3.4

Multiply $2$ by $1$ .

$0 + \sec^{2} (2 x) \cdot 2$

Step 1.1.3.5

Move $2$ to the left of $\sec^{2} (2 x)$ .

$0 + 2 \sec^{2} (2 x)$

Step 1.1.4

Add $0$ and $2 \sec^{2} (2 x)$ .

$2 \sec^{2} (2 x)$

Step 1.2

Substitute the lower limit in for $x$ in $u_{2} = 3 + \tan (2 x)$ .

$u_{lower} = 3 + \tan (2 \cdot 0)$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $2$ by $0$ .

$u_{lower} = 3 + \tan (0)$

Step 1.3.1.2

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

$u_{lower} = 3 + 0$

Step 1.3.2

Add $3$ and $0$ .

$u_{lower} = 3$

Step 1.4

Substitute the upper limit in for $x$ in $u_{2} = 3 + \tan (2 x)$ .

$u_{upper} = 3 + \tan (2 \frac{π}{8})$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Cancel the common factor of $2$ .

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Step 1.5.1.1.1

Factor $2$ out of $8$ .

$u_{upper} = 3 + \tan (2 \frac{π}{2 (4)})$

Step 1.5.1.1.2

Cancel the common factor.

$u_{upper} = 3 + \tan (2 \frac{π}{2 \cdot 4})$

Step 1.5.1.1.3

Rewrite the expression.

$u_{upper} = 3 + \tan (\frac{π}{4})$

Step 1.5.1.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$u_{upper} = 3 + 1$

Step 1.5.2

Add $3$ and $1$ .

$u_{upper} = 4$

Step 1.6

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

$u_{lower} = 3$

$u_{upper} = 4$

Step 1.7

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

$\int_{3}^{4} \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{2}$ .

$\int_{3}^{4} \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 2.2

Move $2$ to the left of $u_{2}$ .

$\int_{3}^{4} \frac{1}{2 u_{2}} d u_{2}$

Step 3

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

$\frac{1}{2} \int_{3}^{4} \frac{1}{u_{2}} d u_{2}$

Step 4

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{1}{2} \ln (| u_{2} |)]_{3}^{4}$

Step 5

Evaluate $\ln (| u_{2} |)$ at $4$ and at $3$ .

$\frac{1}{2} (\ln (| 4 |) - \ln (| 3 |))$

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{2} \ln (\frac{| 4 |}{| 3 |})$

Step 6.2

Combine $\frac{1}{2}$ and $\ln (\frac{| 4 |}{| 3 |})$ .

$\frac{\ln (\frac{| 4 |}{| 3 |})}{2}$

Step 7

Simplify.

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Step 7.1

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{\ln (\frac{4}{| 3 |})}{2}$

Step 7.2

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{\ln (\frac{4}{3})}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\frac{4}{3})}{2}$

Decimal Form:

$0.14384103 \dots$