Calculus Examples

$\int_{\frac{π}{12}}^{\frac{π}{9}} \sec^{2} (3 x) d x$

Step 1

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 1.1.3

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

$3 \cdot 1$

Step 1.1.4

Multiply $3$ by $1$ .

$3$

Step 1.2

Substitute the lower limit in for $x$ in $u = 3 x$ .

$u_{lower} = 3 \frac{π}{12}$

Step 1.3

Cancel the common factor of $3$ .

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Step 1.3.1

Factor $3$ out of $12$ .

$u_{lower} = 3 \frac{π}{3 (4)}$

Step 1.3.2

Cancel the common factor.

$u_{lower} = 3 \frac{π}{3 \cdot 4}$

Step 1.3.3

Rewrite the expression.

$u_{lower} = \frac{π}{4}$

Step 1.4

Substitute the upper limit in for $x$ in $u = 3 x$ .

$u_{upper} = 3 \frac{π}{9}$

Step 1.5

Cancel the common factor of $3$ .

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Step 1.5.1

Factor $3$ out of $9$ .

$u_{upper} = 3 \frac{π}{3 (3)}$

Step 1.5.2

Cancel the common factor.

$u_{upper} = 3 \frac{π}{3 \cdot 3}$

Step 1.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{3}$

Step 1.6

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

$u_{lower} = \frac{π}{4}$

$u_{upper} = \frac{π}{3}$

Step 1.7

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

$\int_{\frac{π}{4}}^{\frac{π}{3}} \sec^{2} (u) \frac{1}{3} d u$

Step 2

Combine $\sec^{2} (u)$ and $\frac{1}{3}$ .

$\int_{\frac{π}{4}}^{\frac{π}{3}} \frac{\sec^{2} (u)}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{\frac{π}{4}}^{\frac{π}{3}} \sec^{2} (u) d u$

Step 4

Since the derivative of $\tan (u)$ is $\sec^{2} (u)$ , the integral of $\sec^{2} (u)$ is $\tan (u)$ .

$\frac{1}{3} \tan (u)]_{\frac{π}{4}}^{\frac{π}{3}}$

Step 5

Evaluate $\tan (u)$ at $\frac{π}{3}$ and at $\frac{π}{4}$ .

$\frac{1}{3} (\tan (\frac{π}{3}) - \tan (\frac{π}{4}))$

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{3} (\sqrt{3} - \tan (\frac{π}{4}))$

Step 6.2

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

$\frac{1}{3} (\sqrt{3} - 1 \cdot 1)$

Step 6.3

Multiply $- 1$ by $1$ .

$\frac{1}{3} (\sqrt{3} - 1)$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1}{3} \sqrt{3} + \frac{1}{3} \cdot - 1$

Step 7.2

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

$\frac{\sqrt{3}}{3} + \frac{1}{3} \cdot - 1$

Step 7.3

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

$\frac{\sqrt{3}}{3} + \frac{- 1}{3}$

Step 7.4

Move the negative in front of the fraction.

$\frac{\sqrt{3}}{3} - \frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{3} - \frac{1}{3}$

Decimal Form:

$0.24401693 \dots$