Calculus Examples

$\sqrt{4 + x^{2}}$

Step 1

Write $\sqrt{4 + x^{2}}$ as a function.

$f (x) = \sqrt{4 + x^{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \sqrt{4 + x^{2}} d x$

Step 4

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

$\int \sqrt{4 + {(2 \tan (t))}^{2}} (2 \sec^{2} (t)) d t$

Step 5

Simplify terms.

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Step 5.1

Simplify $\sqrt{4 + {(2 \tan (t))}^{2}}$ .

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the product rule to $2 \tan (t)$ .

$\int \sqrt{4 + 2^{2} \tan^{2} (t)} (2 \sec^{2} (t)) d t$

Step 5.1.1.2

Raise $2$ to the power of $2$ .

$\int \sqrt{4 + 4 \tan^{2} (t)} (2 \sec^{2} (t)) d t$

Step 5.1.2

Factor $4$ out of $4$ .

$\int \sqrt{4 (1) + 4 \tan^{2} (t)} (2 \sec^{2} (t)) d t$

Step 5.1.3

Factor $4$ out of $4 \tan^{2} (t)$ .

$\int \sqrt{4 (1) + 4 (\tan^{2} (t))} (2 \sec^{2} (t)) d t$

Step 5.1.4

Factor $4$ out of $4 (1) + 4 (\tan^{2} (t))$ .

$\int \sqrt{4 (1 + \tan^{2} (t))} (2 \sec^{2} (t)) d t$

Step 5.1.5

Rearrange terms.

$\int \sqrt{4 (\tan^{2} (t) + 1)} (2 \sec^{2} (t)) d t$

Step 5.1.6

Apply pythagorean identity.

$\int \sqrt{4 \sec^{2} (t)} (2 \sec^{2} (t)) d t$

Step 5.1.7

Rewrite $4 \sec^{2} (t)$ as ${(2 \sec (t))}^{2}$ .

$\int \sqrt{{(2 \sec (t))}^{2}} (2 \sec^{2} (t)) d t$

Step 5.1.8

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

$\int 2 \sec (t) (2 \sec^{2} (t)) d t$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $2$ by $2$ .

$\int 4 \sec (t) \sec^{2} (t) d t$

Step 5.2.2

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

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Step 5.2.2.1

Move $\sec^{2} (t)$ .

$\int 4 (\sec^{2} (t) \sec (t)) d t$

Step 5.2.2.2

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

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Step 5.2.2.2.1

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

$\int 4 (\sec^{2} (t) \sec^{1} (t)) d t$

Step 5.2.2.2.2

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

$\int 4 {\sec (t)}^{2 + 1} d t$

Step 5.2.2.3

Add $2$ and $1$ .

$\int 4 \sec^{3} (t) d t$

Step 6

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$4 \int \sec^{3} (t) d t$

Step 7

Factor $\sec (t)$ out of $\sec^{3} (t)$ .

$4 \int \sec (t) \sec^{2} (t) d t$

Step 8

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

$4 (\sec (t) \tan (t) - \int \tan (t) (\sec (t) \tan (t)) d t)$

Step 9

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

$4 (\sec (t) \tan (t) - \int \tan^{1} (t) \tan (t) \sec (t) d t)$

Step 10

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

$4 (\sec (t) \tan (t) - \int \tan^{1} (t) \tan^{1} (t) \sec (t) d t)$

Step 11

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

$4 (\sec (t) \tan (t) - \int {\tan (t)}^{1 + 1} \sec (t) d t)$

Step 12

Simplify the expression.

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Step 12.1

Add $1$ and $1$ .

$4 (\sec (t) \tan (t) - \int \tan^{2} (t) \sec (t) d t)$

Step 12.2

Reorder $\tan^{2} (t)$ and $\sec (t)$ .

$4 (\sec (t) \tan (t) - \int \sec (t) \tan^{2} (t) d t)$

Step 13

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

$4 (\sec (t) \tan (t) - \int \sec (t) (- 1 + \sec^{2} (t)) d t)$

Step 14

Simplify by multiplying through.

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Step 14.1

Rewrite the exponentiation as a product.

$4 (\sec (t) \tan (t) - \int \sec (t) (- 1 + \sec (t) \sec (t)) d t)$

Step 14.2

Apply the distributive property.

$4 (\sec (t) \tan (t) - \int \sec (t) \cdot - 1 + \sec (t) (\sec (t) \sec (t)) d t)$

Step 14.3

Reorder $\sec (t)$ and $- 1$ .

$4 (\sec (t) \tan (t) - \int - 1 \cdot \sec (t) + \sec (t) (\sec (t) \sec (t)) d t)$

Step 15

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

$4 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec (t) \sec (t) d t)$

Step 16

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

$4 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec^{1} (t) \sec (t) d t)$

Step 17

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

$4 (\sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{1 + 1} \sec (t) d t)$

Step 18

Add $1$ and $1$ .

$4 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec (t) d t)$

Step 19

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

$4 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec^{1} (t) d t)$

Step 20

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

$4 (\sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{2 + 1} d t)$

Step 21

Add $2$ and $1$ .

$4 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{3} (t) d t)$

Step 22

Split the single integral into multiple integrals.

$4 (\sec (t) \tan (t) - (\int - 1 \sec (t) d t + \int \sec^{3} (t) d t))$

Step 23

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$4 (\sec (t) \tan (t) - (- \int \sec (t) d t + \int \sec^{3} (t) d t))$

Step 24

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

$4 (\sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t))$

Step 25

Simplify by multiplying through.

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Step 25.1

Apply the distributive property.

$4 (\sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 25.2

Multiply $- 1$ by $- 1$ .

$4 (\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 26

Solving for $\int \sec^{3} (t) d t$ , we find that $\int \sec^{3} (t) d t$ = $\frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2}$ .

$4 (\frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2} + C)$

Step 27

Multiply $\ln (| \sec (t) + \tan (t) |) + C$ by $1$ .

$4 (\frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C}{2} + C)$

Step 28

Simplify.

$4 (\frac{1}{2}) (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 29

Simplify.

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Step 29.1

Combine $4$ and $\frac{1}{2}$ .

$\frac{4}{2} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 29.2

Cancel the common factor of $4$ and $2$ .

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Step 29.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 29.2.2

Cancel the common factors.

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Step 29.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 29.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 29.2.2.3

Rewrite the expression.

$\frac{2}{1} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 29.2.2.4

Divide $2$ by $1$ .

$2 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 30

Replace all occurrences of $t$ with $\arctan (\frac{x}{2})$ .

$2 (\sec (\arctan (\frac{x}{2})) \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31

Simplify.

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Step 31.1

Simplify each term.

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Step 31.1.1

Draw a triangle in the plane with vertices $(1, \frac{x}{2})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{x}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{x}{2})$ . Therefore, $\sec (\arctan (\frac{x}{2}))$ is $\sqrt{1 + {(\frac{x}{2})}^{2}}$ .

$2 (\sqrt{1 + {(\frac{x}{2})}^{2}} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.2

Apply the product rule to $\frac{x}{2}$ .

$2 (\sqrt{1 + \frac{x^{2}}{2^{2}}} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.3

Raise $2$ to the power of $2$ .

$2 (\sqrt{1 + \frac{x^{2}}{4}} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.4

Write $1$ as a fraction with a common denominator.

$2 (\sqrt{\frac{4}{4} + \frac{x^{2}}{4}} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.5

Combine the numerators over the common denominator.

$2 (\sqrt{\frac{4 + x^{2}}{4}} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.6

Rewrite $\frac{4 + x^{2}}{4}$ as ${(\frac{1}{2})}^{2} (4 + x^{2})$ .

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Step 31.1.6.1

Factor the perfect power $1^{2}$ out of $4 + x^{2}$ .

$2 (\sqrt{\frac{1^{2} (4 + x^{2})}{4}} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.6.2

Factor the perfect power $2^{2}$ out of $4$ .

$2 (\sqrt{\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.6.3

Rearrange the fraction $\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}$ .

$2 (\sqrt{{(\frac{1}{2})}^{2} (4 + x^{2})} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.7

Pull terms out from under the radical.

$2 (\frac{1}{2} \sqrt{4 + x^{2}} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.8

Combine $\frac{1}{2}$ and $\sqrt{4 + x^{2}}$ .

$2 (\frac{\sqrt{4 + x^{2}}}{2} \tan (\arctan (\frac{x}{2})) + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.9

The functions tangent and arctangent are inverses.

$2 (\frac{\sqrt{4 + x^{2}}}{2} \cdot \frac{x}{2} + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.10

Combine.

$2 (\frac{\sqrt{4 + x^{2}} x}{2 \cdot 2} + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.11

Multiply $2$ by $2$ .

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12

Simplify each term.

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Step 31.1.12.1

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \sqrt{1 + {(\frac{x}{2})}^{2}} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.2

Apply the product rule to $\frac{x}{2}$ .

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \sqrt{1 + \frac{x^{2}}{2^{2}}} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.3

Raise $2$ to the power of $2$ .

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \sqrt{1 + \frac{x^{2}}{4}} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.4

Write $1$ as a fraction with a common denominator.

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \sqrt{\frac{4}{4} + \frac{x^{2}}{4}} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.5

Combine the numerators over the common denominator.

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \sqrt{\frac{4 + x^{2}}{4}} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.6

Rewrite $\frac{4 + x^{2}}{4}$ as ${(\frac{1}{2})}^{2} (4 + x^{2})$ .

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Step 31.1.12.6.1

Factor the perfect power $1^{2}$ out of $4 + x^{2}$ .

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \sqrt{\frac{1^{2} (4 + x^{2})}{4}} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.6.2

Factor the perfect power $2^{2}$ out of $4$ .

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \sqrt{\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.6.3

Rearrange the fraction $\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}$ .

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \sqrt{{(\frac{1}{2})}^{2} (4 + x^{2})} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.7

Pull terms out from under the radical.

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \frac{1}{2} \sqrt{4 + x^{2}} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.8

Combine $\frac{1}{2}$ and $\sqrt{4 + x^{2}}$ .

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \frac{\sqrt{4 + x^{2}}}{2} + \tan (\arctan (\frac{x}{2})) |)) + C$

Step 31.1.12.9

The functions tangent and arctangent are inverses.

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \frac{\sqrt{4 + x^{2}}}{2} + \frac{x}{2} |)) + C$

Step 31.1.13

Combine the numerators over the common denominator.

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (| \frac{\sqrt{4 + x^{2}} + x}{2} |)) + C$

Step 31.1.14

Remove non-negative terms from the absolute value.

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (\frac{| \sqrt{4 + x^{2}} + x |}{2})) + C$

Step 31.2

To write $\ln (\frac{| \sqrt{4 + x^{2}} + x |}{2})$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \ln (\frac{| \sqrt{4 + x^{2}} + x |}{2}) \cdot \frac{4}{4}) + C$

Step 31.3

Combine $\ln (\frac{| \sqrt{4 + x^{2}} + x |}{2})$ and $\frac{4}{4}$ .

$2 (\frac{\sqrt{4 + x^{2}} x}{4} + \frac{\ln (\frac{| \sqrt{4 + x^{2}} + x |}{2}) \cdot 4}{4}) + C$

Step 31.4

Combine the numerators over the common denominator.

$2 \frac{\sqrt{4 + x^{2}} x + \ln (\frac{| \sqrt{4 + x^{2}} + x |}{2}) \cdot 4}{4} + C$

Step 31.5

Move $4$ to the left of $\ln (\frac{| \sqrt{4 + x^{2}} + x |}{2})$ .

$2 \frac{\sqrt{4 + x^{2}} x + 4 \ln (\frac{| \sqrt{4 + x^{2}} + x |}{2})}{4} + C$

Step 31.6

Cancel the common factor of $2$ .

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Step 31.6.1

Factor $2$ out of $4$ .

$2 \frac{\sqrt{4 + x^{2}} x + 4 \ln (\frac{| \sqrt{4 + x^{2}} + x |}{2})}{2 (2)} + C$

Step 31.6.2

Cancel the common factor.

$2 \frac{\sqrt{4 + x^{2}} x + 4 \ln (\frac{| \sqrt{4 + x^{2}} + x |}{2})}{2 \cdot 2} + C$

Step 31.6.3

Rewrite the expression.

$\frac{\sqrt{4 + x^{2}} x + 4 \ln (\frac{| \sqrt{4 + x^{2}} + x |}{2})}{2} + C$

Step 32

Reorder terms.

$\frac{1}{2} (\sqrt{4 + x^{2}} x + 4 \ln (\frac{1}{2} | \sqrt{4 + x^{2}} + x |)) + C$

Step 33

The answer is the antiderivative of the function $f (x) = \sqrt{4 + x^{2}}$ .

$F (x) =$ $\frac{1}{2} (\sqrt{4 + x^{2}} x + 4 \ln (\frac{1}{2} | \sqrt{4 + x^{2}} + x |)) + C$