Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\frac{x}{3}$ .

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

Step 1.1.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$\frac{1}{3} \frac{d}{d x} [x]$

Step 1.1.3

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

$\frac{1}{3} \cdot 1$

Step 1.1.4

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{x}{3}$ .

$u_{lower} = \frac{0}{3}$

Step 1.3

Divide $0$ by $3$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{x}{3}$ .

$u_{upper} = \frac{π}{3}$

Step 1.5

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{3}$

Step 1.6

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

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

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

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

Step 2.2

Multiply $3$ by $1$ .

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

Step 2.3

Move $3$ to the left of $\tan^{2} (u)$ .

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

Step 3

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

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

Step 4

Using the Pythagorean Identity, rewrite $\tan^{2} (u)$ as $- 1 + \sec^{2} (u)$ .

$3 \int_{0}^{\frac{π}{3}} - 1 + \sec^{2} (u) d u$

Step 5

Split the single integral into multiple integrals.

$3 (\int_{0}^{\frac{π}{3}} - 1 d u + \int_{0}^{\frac{π}{3}} \sec^{2} (u) d u)$

Step 6

Apply the constant rule.

$3 (- u]_{0}^{\frac{π}{3}} + \int_{0}^{\frac{π}{3}} \sec^{2} (u) d u)$

Step 7

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

$3 (- u]_{0}^{\frac{π}{3}} + \tan (u)]_{0}^{\frac{π}{3}})$

Step 8

Combine $- u]_{0}^{\frac{π}{3}}$ and $\tan (u)]_{0}^{\frac{π}{3}}$ .

$3 (- u + \tan (u)]_{0}^{\frac{π}{3}})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $- u + \tan (u)$ at $\frac{π}{3}$ and at $0$ .

$3 ((- \frac{π}{3} + \tan (\frac{π}{3})) - (- 0 + \tan (0)))$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $- 1$ by $0$ .

$3 (- \frac{π}{3} + \tan (\frac{π}{3}) - (0 + \tan (0)))$

Step 9.2.2

Add $0$ and $\tan (0)$ .

$3 (- \frac{π}{3} + \tan (\frac{π}{3}) - \tan (0))$

Step 10

Simplify.

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Step 10.1

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

$3 (- \frac{π}{3} + \sqrt{3} - \tan (0))$

Step 10.2

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

$3 (- \frac{π}{3} + \sqrt{3} - 0)$

Step 10.3

Multiply $- 1$ by $0$ .

$3 (- \frac{π}{3} + \sqrt{3} + 0)$

Step 10.4

Add $- \frac{π}{3} + \sqrt{3}$ and $0$ .

$3 (- \frac{π}{3} + \sqrt{3})$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$3 (- \frac{π}{3}) + 3 \sqrt{3}$

Step 11.2

Cancel the common factor of $3$ .

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Step 11.2.1

Move the leading negative in $- \frac{π}{3}$ into the numerator.

$3 \frac{- π}{3} + 3 \sqrt{3}$

Step 11.2.2

Cancel the common factor.

$3 \frac{- π}{3} + 3 \sqrt{3}$

Step 11.2.3

Rewrite the expression.

$- π + 3 \sqrt{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- π + 3 \sqrt{3}$

Decimal Form:

$2.05455976 \dots$