Calculus Examples

$\int_{0}^{\frac{π}{4}} x \sec^{2} (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sec^{2} (x)$ .

$x \tan (x)]_{0}^{\frac{π}{4}} - \int_{0}^{\frac{π}{4}} \tan (x) d x$

Step 2

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$x \tan (x)]_{0}^{\frac{π}{4}} - (\ln (| \sec (x) |)]_{0}^{\frac{π}{4}})$

Step 3

Simplify the answer.

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Step 3.1

Substitute and simplify.

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Step 3.1.1

Evaluate $x \tan (x)$ at $\frac{π}{4}$ and at $0$ .

$(\frac{π}{4} \tan (\frac{π}{4})) + 0 \tan (0) - (\ln (| \sec (x) |)]_{0}^{\frac{π}{4}})$

Step 3.1.2

Evaluate $\ln (| \sec (x) |)$ at $\frac{π}{4}$ and at $0$ .

$(\frac{π}{4} \tan (\frac{π}{4})) + 0 \tan (0) - ((\ln (| \sec (\frac{π}{4}) |)) - \ln (| \sec (0) |))$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Combine $\frac{π}{4}$ and $\tan (\frac{π}{4})$ .

$\frac{π \tan (\frac{π}{4})}{4} + 0 \tan (0) - ((\ln (| \sec (\frac{π}{4}) |)) - \ln (| \sec (0) |))$

Step 3.1.3.2

Multiply $0$ by $\tan (0)$ .

$\frac{π \tan (\frac{π}{4})}{4} + 0 - ((\ln (| \sec (\frac{π}{4}) |)) - \ln (| \sec (0) |))$

Step 3.1.3.3

Add $\frac{π \tan (\frac{π}{4})}{4}$ and $0$ .

$\frac{π \tan (\frac{π}{4})}{4} - ((\ln (| \sec (\frac{π}{4}) |)) - \ln (| \sec (0) |))$

Step 3.1.3.4

To write $- (\ln (| \sec (\frac{π}{4}) |) - \ln (| \sec (0) |))$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{π \tan (\frac{π}{4})}{4} - (\ln (| \sec (\frac{π}{4}) |) - \ln (| \sec (0) |)) \cdot \frac{4}{4}$

Step 3.1.3.5

Combine $- (\ln (| \sec (\frac{π}{4}) |) - \ln (| \sec (0) |))$ and $\frac{4}{4}$ .

$\frac{π \tan (\frac{π}{4})}{4} + \frac{- (\ln (| \sec (\frac{π}{4}) |) - \ln (| \sec (0) |)) \cdot 4}{4}$

Step 3.1.3.6

Combine the numerators over the common denominator.

$\frac{π \tan (\frac{π}{4}) - (\ln (| \sec (\frac{π}{4}) |) - \ln (| \sec (0) |)) \cdot 4}{4}$

Step 3.1.3.7

Multiply $4$ by $- 1$ .

$\frac{π \tan (\frac{π}{4}) - 4 (\ln (| \sec (\frac{π}{4}) |) - \ln (| \sec (0) |))}{4}$

Step 3.2

Simplify.

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Step 3.2.1

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

$\frac{π \cdot 1 - 4 (\ln (| \sec (\frac{π}{4}) |) - \ln (| \sec (0) |))}{4}$

Step 3.2.2

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\frac{π \cdot 1 - 4 (\ln (| \frac{2}{\sqrt{2}} |) - \ln (| \sec (0) |))}{4}$

Step 3.2.3

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

$\frac{π \cdot 1 - 4 (\ln (| \frac{2}{\sqrt{2}} |) - \ln (| 1 |))}{4}$

Step 3.2.4

Multiply $π$ by $1$ .

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

Step 3.2.5

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

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

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{π - 4 \ln (\frac{| \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} |}{| 1 |})}{4}$

Step 3.3.1.2

Combine and simplify the denominator.

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Step 3.3.1.2.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.3.1.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{π - 4 \ln (\frac{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} |}{| 1 |})}{4}$

Step 3.3.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{π - 4 \ln (\frac{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} |}{| 1 |})}{4}$

Step 3.3.1.2.4

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

$\frac{π - 4 \ln (\frac{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} |}{| 1 |})}{4}$

Step 3.3.1.2.5

Add $1$ and $1$ .

$\frac{π - 4 \ln (\frac{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} |}{| 1 |})}{4}$

Step 3.3.1.2.6

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

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Step 3.3.1.2.6.1

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

$\frac{π - 4 \ln (\frac{| \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} |}{| 1 |})}{4}$

Step 3.3.1.2.6.2

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

$\frac{π - 4 \ln (\frac{| \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} |}{| 1 |})}{4}$

Step 3.3.1.2.6.3

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

$\frac{π - 4 \ln (\frac{| \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} |}{| 1 |})}{4}$

Step 3.3.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.3.1.2.6.4.1

Cancel the common factor.

$\frac{π - 4 \ln (\frac{| \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} |}{| 1 |})}{4}$

Step 3.3.1.2.6.4.2

Rewrite the expression.

$\frac{π - 4 \ln (\frac{| \frac{2 \sqrt{2}}{2^{1}} |}{| 1 |})}{4}$

Step 3.3.1.2.6.5

Evaluate the exponent.

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

Step 3.3.1.3

Cancel the common factor of $2$ .

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Step 3.3.1.3.1

Cancel the common factor.

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

Step 3.3.1.3.2

Divide $\sqrt{2}$ by $1$ .

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

Step 3.3.1.4

$\sqrt{2}$ is approximately $1.41421356$ which is positive so remove the absolute value

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

Step 3.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π - 4 \ln (\frac{\sqrt{2}}{1})}{4}$

Step 3.3.3

Divide $\sqrt{2}$ by $1$ .

$\frac{π - 4 \ln (\sqrt{2})}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{π - 4 \ln (\sqrt{2})}{4}$

Decimal Form:

$0.43882457 \dots$