Calculus Examples

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

Step 1

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

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

$- θ]_{0}^{\frac{π}{2}} + \tan (θ)]_{0}^{\frac{π}{2}}$

Step 5

Simplify the answer.

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Step 5.1

Combine $- θ]_{0}^{\frac{π}{2}}$ and $\tan (θ)]_{0}^{\frac{π}{2}}$ .

$- θ + \tan (θ)]_{0}^{\frac{π}{2}}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $- θ + \tan (θ)$ at $\frac{π}{2}$ and at $0$ .

$(- \frac{π}{2} + \tan (\frac{π}{2})) - (- 0 + \tan (0))$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $- 1$ by $0$ .

$- \frac{π}{2} + \tan (\frac{π}{2}) - (0 + \tan (0))$

Step 5.2.2.2

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

$- \frac{π}{2} + \tan (\frac{π}{2}) - \tan (0)$

Step 5.3

Simplify.

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Step 5.3.1

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

$- \frac{π}{2} + \tan (\frac{π}{2}) - 0$

Step 5.3.2

Multiply $- 1$ by $0$ .

$- \frac{π}{2} + \tan (\frac{π}{2}) + 0$

Step 5.3.3

Add $- \frac{π}{2} + \tan (\frac{π}{2})$ and $0$ .

$- \frac{π}{2} + \tan (\frac{π}{2})$

Step 6

The exact value of $\tan (\frac{π}{2})$ is $\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0}$ .

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Step 6.1

Split $\frac{π}{2}$ into two angles where the values of the six trigonometric functions are known.

$- \frac{π}{2} + \tan (\frac{π}{6} + \frac{π}{3})$

Step 6.2

Apply the sum of angles identity.

$- \frac{π}{2} + \frac{\tan (\frac{π}{6}) + \tan (\frac{π}{3})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{3})}$

Step 6.3

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \tan (\frac{π}{3})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{3})}$

Step 6.4

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

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{3})}$

Step 6.5

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \tan (\frac{π}{3})}$

Step 6.6

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

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \sqrt{3}}$

Step 6.7

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

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Step 6.7.1

Simplify each term.

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Step 6.7.1.1

Multiply $- \frac{\sqrt{3}}{3} \sqrt{3}$ .

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Step 6.7.1.1.1

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

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3} \sqrt{3}}{3}}$

Step 6.7.1.1.2

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} \sqrt{3}}{3}}$

Step 6.7.1.1.3

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{3}}$

Step 6.7.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1 + 1}}{3}}$

Step 6.7.1.1.5

Add $1$ and $1$ .

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{2}}{3}}$

Step 6.7.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.7.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{(3^{\frac{1}{2}})}^{2}}{3}}$

Step 6.7.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{1}{2} \cdot 2}}{3}}$

Step 6.7.1.2.3

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

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}}$

Step 6.7.1.2.4

Cancel the common factor of $2$ .

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Step 6.7.1.2.4.1

Cancel the common factor.

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}}$

Step 6.7.1.2.4.2

Rewrite the expression.

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{1}}{3}}$

Step 6.7.1.2.5

Evaluate the exponent.

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}}$

Step 6.7.1.3

Cancel the common factor of $3$ .

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Step 6.7.1.3.1

Cancel the common factor.

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}}$

Step 6.7.1.3.2

Rewrite the expression.

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1 \cdot 1}$

Step 6.7.1.4

Multiply $- 1$ by $1$ .

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1}$

Step 6.7.2

Subtract $1$ from $1$ .

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0}$

Step 6.7.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0}$

Step 6.8

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{π}{2} + \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0}$

Step 7

The expression contains a division by $0$ . The expression is undefined.

Undefined