Calculus Examples

$\int_{0}^{1} \sqrt{x^{2} + 1} d x$

Step 1

Let $x = \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \sec^{2} (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\sec^{2} (t)$ is positive.

$\int_{0}^{\frac{π}{4}} \sqrt{\tan^{2} (t) + 1} \sec^{2} (t) d t$

Step 2

Simplify $\sqrt{\tan^{2} (t) + 1}$ .

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Step 2.1

Apply pythagorean identity.

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

Step 2.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 3

Multiply $\sec (t)$ by $\sec^{2} (t)$ by adding the exponents.

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Step 3.1

Multiply $\sec (t)$ by $\sec^{2} (t)$ .

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Step 3.1.1

Raise $\sec (t)$ to the power of $1$ .

$\int_{0}^{\frac{π}{4}} \sec^{1} (t) \sec^{2} (t) d t$

Step 3.1.2

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

$\int_{0}^{\frac{π}{4}} {\sec (t)}^{1 + 2} d t$

Step 3.2

Add $1$ and $2$ .

$\int_{0}^{\frac{π}{4}} \sec^{3} (t) d t$

Step 4

Apply the reduction formula.

$\frac{\tan (t) \sec (t)}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} \int_{0}^{\frac{π}{4}} \sec (t) d t$

Step 5

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

$\frac{\tan (t) \sec (t)}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} \ln (| \sec (t) + \tan (t) |)]_{0}^{\frac{π}{4}}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{\tan (t) \sec (t)}{2}$ at $\frac{π}{4}$ and at $0$ .

$(\frac{\tan (\frac{π}{4}) \sec (\frac{π}{4})}{2}) - \frac{\tan (0) \sec (0)}{2} + \frac{1}{2} \ln (| \sec (t) + \tan (t) |)]_{0}^{\frac{π}{4}}$

Step 6.2

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

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

Step 6.3

Remove parentheses.

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

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

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

Step 7.7

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

$\frac{1 \frac{2}{\sqrt{2}}}{2} - \frac{0 \cdot 1}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + \tan (0) |))$

Step 7.8

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

$\frac{1 \frac{2}{\sqrt{2}}}{2} - \frac{0 \cdot 1}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.9

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

$\frac{\frac{2}{\sqrt{2}}}{2} - \frac{0 \cdot 1}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.10

Rewrite $\frac{\frac{2}{\sqrt{2}}}{2}$ as a product.

$\frac{2}{\sqrt{2}} \cdot \frac{1}{2} - \frac{0 \cdot 1}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.11

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

$\frac{2}{\sqrt{2} \cdot 2} - \frac{0 \cdot 1}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.12

Move $2$ to the left of $\sqrt{2}$ .

$\frac{2}{2 \cdot \sqrt{2}} - \frac{0 \cdot 1}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.13

Cancel the common factor of $2$ .

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Step 7.13.1

Cancel the common factor.

$\frac{2}{2 \sqrt{2}} - \frac{0 \cdot 1}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.13.2

Rewrite the expression.

$\frac{1}{\sqrt{2}} - \frac{0 \cdot 1}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.14

Multiply $0$ by $1$ .

$\frac{1}{\sqrt{2}} - \frac{0}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.15

Cancel the common factor of $0$ and $2$ .

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Step 7.15.1

Factor $2$ out of $0$ .

$\frac{1}{\sqrt{2}} - \frac{2 (0)}{2} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.15.2

Cancel the common factors.

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Step 7.15.2.1

Factor $2$ out of $2$ .

$\frac{1}{\sqrt{2}} - \frac{2 \cdot 0}{2 \cdot 1} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.15.2.2

Cancel the common factor.

$\frac{1}{\sqrt{2}} - \frac{2 \cdot 0}{2 \cdot 1} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.15.2.3

Rewrite the expression.

$\frac{1}{\sqrt{2}} - \frac{0}{1} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.15.2.4

Divide $0$ by $1$ .

$\frac{1}{\sqrt{2}} - 0 + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.16

Multiply $- 1$ by $0$ .

$\frac{1}{\sqrt{2}} + 0 + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.17

Add $\frac{1}{\sqrt{2}}$ and $0$ .

$\frac{1}{\sqrt{2}} + \frac{1}{2} (\ln (| \frac{2}{\sqrt{2}} + 1 |) - \ln (| 1 + 0 |))$

Step 7.18

Add $1$ and $0$ .

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

Step 7.19

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

$\frac{1}{\sqrt{2}} + \frac{1}{2} \ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |})$

Step 7.20

Combine $\frac{1}{2}$ and $\ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |})$ .

$\frac{1}{\sqrt{2}} + \frac{\ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |})}{2}$

Step 7.21

To write $\frac{1}{\sqrt{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{\sqrt{2}} \cdot \frac{2}{2} + \frac{\ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |})}{2}$

Step 7.22

To write $\frac{\ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |})}{2}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{\sqrt{2}} \cdot \frac{2}{2} + \frac{\ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |})}{2} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 7.23

Write each expression with a common denominator of $\sqrt{2} \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.23.1

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

$\frac{2}{\sqrt{2} \cdot 2} + \frac{\ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |})}{2} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 7.23.2

Multiply $\frac{\ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |})}{2}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2}{\sqrt{2} \cdot 2} + \frac{\ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 7.23.3

Reorder the factors of $\sqrt{2} \cdot 2$ .

$\frac{2}{2 \sqrt{2}} + \frac{\ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 7.24

Combine the numerators over the common denominator.

$\frac{2 + \ln (\frac{| \frac{2}{\sqrt{2}} + 1 |}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

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

$\frac{2 + \ln (\frac{| \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + 1 |}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 8.1.1.2

Combine and simplify the denominator.

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Step 8.1.1.2.1

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

$\frac{2 + \ln (\frac{| \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} + 1 |}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 8.1.1.2.2

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

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

Step 8.1.1.2.3

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

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

Step 8.1.1.2.4

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

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

Step 8.1.1.2.5

Add $1$ and $1$ .

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

Step 8.1.1.2.6

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

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Step 8.1.1.2.6.1

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

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

Step 8.1.1.2.6.2

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

$\frac{2 + \ln (\frac{| \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + 1 |}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 8.1.1.2.6.3

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

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

Step 8.1.1.2.6.4

Cancel the common factor of $2$ .

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Step 8.1.1.2.6.4.1

Cancel the common factor.

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

Step 8.1.1.2.6.4.2

Rewrite the expression.

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

Step 8.1.1.2.6.5

Evaluate the exponent.

$\frac{2 + \ln (\frac{| \frac{2 \sqrt{2}}{2} + 1 |}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 8.1.1.3

Cancel the common factor of $2$ .

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Step 8.1.1.3.1

Cancel the common factor.

$\frac{2 + \ln (\frac{| \frac{2 \sqrt{2}}{2} + 1 |}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 8.1.1.3.2

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

$\frac{2 + \ln (\frac{| \sqrt{2} + 1 |}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 8.1.2

$\sqrt{2} + 1$ is approximately $2.41421356$ which is positive so remove the absolute value

$\frac{2 + \ln (\frac{\sqrt{2} + 1}{| 1 |}) \sqrt{2}}{2 \sqrt{2}}$

Step 8.2

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

$\frac{2 + \ln (\frac{\sqrt{2} + 1}{1}) \sqrt{2}}{2 \sqrt{2}}$

Step 8.3

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

$\frac{2 + \ln (\sqrt{2} + 1) \sqrt{2}}{2 \sqrt{2}}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{2 + \ln (\sqrt{2} + 1) \sqrt{2}}{2 \sqrt{2}}$

Decimal Form:

$1.14779357 \dots$

Step 10