Calculus Examples

$\int \sqrt{x^{2} + 8 x + 6} d x$

Step 1

Complete the square.

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Step 1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = 8$

$c = 6$

Step 1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{8}{2 \cdot 1}$

Step 1.3.2

Cancel the common factor of $8$ and $2$ .

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Step 1.3.2.1

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2 \cdot 1}$

Step 1.3.2.2

Cancel the common factors.

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Step 1.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot 4}{2 (1)}$

Step 1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot 1}$

Step 1.3.2.2.3

Rewrite the expression.

$d = \frac{4}{1}$

Step 1.3.2.2.4

Divide $4$ by $1$ .

$d = 4$

Step 1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 6 - \frac{8^{2}}{4 \cdot 1}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Raise $8$ to the power of $2$ .

$e = 6 - \frac{64}{4 \cdot 1}$

Step 1.4.2.1.2

Multiply $4$ by $1$ .

$e = 6 - \frac{64}{4}$

Step 1.4.2.1.3

Divide $64$ by $4$ .

$e = 6 - 1 \cdot 16$

Step 1.4.2.1.4

Multiply $- 1$ by $16$ .

$e = 6 - 16$

Step 1.4.2.2

Subtract $16$ from $6$ .

$e = - 10$

Step 1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 4)}^{2} - 10$ .

$\int \sqrt{{(x + 4)}^{2} - 10} d x$

Step 2

Let $u = x + 4$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x + 4$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x + 4$ .

$\frac{d}{d x} [x + 4]$

Step 2.1.2

By the Sum Rule, the derivative of $x + 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [4]$

Step 2.1.3

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

$1 + \frac{d}{d x} [4]$

Step 2.1.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

Rewrite the problem using $u$ and $d u$ .

$\int \sqrt{u^{2} - 10} d u$

Step 3

Let $u = \sqrt{10} \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u = \sqrt{10} \sec (t) \tan (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\sqrt{10} \sec (t) \tan (t)$ is positive.

$\int \sqrt{{(\sqrt{10} \sec (t))}^{2} - 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4

Simplify terms.

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Step 4.1

Simplify $\sqrt{{(\sqrt{10} \sec (t))}^{2} - 10}$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $\sqrt{10} \sec (t)$ .

$\int \sqrt{{\sqrt{10}}^{2} \sec^{2} (t) - 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.1.2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 4.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\int \sqrt{{(10^{\frac{1}{2}})}^{2} \sec^{2} (t) - 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.1.2.2

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

$\int \sqrt{10^{\frac{1}{2} \cdot 2} \sec^{2} (t) - 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.1.2.3

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

$\int \sqrt{10^{\frac{2}{2}} \sec^{2} (t) - 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.1.2.4.1

Cancel the common factor.

$\int \sqrt{10^{\frac{2}{2}} \sec^{2} (t) - 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.1.2.4.2

Rewrite the expression.

$\int \sqrt{10^{1} \sec^{2} (t) - 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.1.2.5

Evaluate the exponent.

$\int \sqrt{10 \sec^{2} (t) - 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.2

Factor $10$ out of $10 \sec^{2} (t)$ .

$\int \sqrt{10 (\sec^{2} (t)) - 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.3

Factor $10$ out of $- 10$ .

$\int \sqrt{10 \sec^{2} (t) + 10 \cdot - 1} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.4

Factor $10$ out of $10 \sec^{2} (t) + 10 \cdot - 1$ .

$\int \sqrt{10 (\sec^{2} (t) - 1)} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.5

Apply pythagorean identity.

$\int \sqrt{10 \tan^{2} (t)} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.6

Reorder $10$ and $\tan^{2} (t)$ .

$\int \sqrt{\tan^{2} (t) \cdot 10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.1.7

Pull terms out from under the radical.

$\int \tan (t) \sqrt{10} (\sqrt{10} \sec (t) \tan (t)) d t$

Step 4.2

Simplify.

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Step 4.2.1

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

$\int \tan^{1} (t) \tan (t) \sqrt{10} (\sqrt{10} \sec (t)) d t$

Step 4.2.2

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

$\int \tan^{1} (t) \tan^{1} (t) \sqrt{10} (\sqrt{10} \sec (t)) d t$

Step 4.2.3

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

$\int {\tan (t)}^{1 + 1} \sqrt{10} (\sqrt{10} \sec (t)) d t$

Step 4.2.4

Add $1$ and $1$ .

$\int \tan^{2} (t) \sqrt{10} (\sqrt{10} \sec (t)) d t$

Step 4.2.5

Raise $\sqrt{10}$ to the power of $1$ .

$\int \tan^{2} (t) ({\sqrt{10}}^{1} \sqrt{10}) \sec (t) d t$

Step 4.2.6

Raise $\sqrt{10}$ to the power of $1$ .

$\int \tan^{2} (t) ({\sqrt{10}}^{1} {\sqrt{10}}^{1}) \sec (t) d t$

Step 4.2.7

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

$\int \tan^{2} (t) {\sqrt{10}}^{1 + 1} \sec (t) d t$

Step 4.2.8

Add $1$ and $1$ .

$\int \tan^{2} (t) {\sqrt{10}}^{2} \sec (t) d t$

Step 4.2.9

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 4.2.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\int \tan^{2} (t) {(10^{\frac{1}{2}})}^{2} \sec (t) d t$

Step 4.2.9.2

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

$\int \tan^{2} (t) \cdot 10^{\frac{1}{2} \cdot 2} \sec (t) d t$

Step 4.2.9.3

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

$\int \tan^{2} (t) \cdot 10^{\frac{2}{2}} \sec (t) d t$

Step 4.2.9.4

Cancel the common factor of $2$ .

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Step 4.2.9.4.1

Cancel the common factor.

$\int \tan^{2} (t) \cdot 10^{\frac{2}{2}} \sec (t) d t$

Step 4.2.9.4.2

Rewrite the expression.

$\int \tan^{2} (t) \cdot 10^{1} \sec (t) d t$

Step 4.2.9.5

Evaluate the exponent.

$\int \tan^{2} (t) \cdot 10 \sec (t) d t$

Step 4.2.10

Move $10$ to the left of $\tan^{2} (t)$ .

$\int 10 \tan^{2} (t) \sec (t) d t$

Step 5

Since $10$ is constant with respect to $t$ , move $10$ out of the integral.

$10 \int \tan^{2} (t) \sec (t) d t$

Step 6

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

$10 \int \tan^{2} (t) \sec^{1} (t) d t$

Step 7

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

$10 \int (- 1 + \sec^{2} (t)) \sec^{1} (t) d t$

Step 8

Simplify terms.

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Step 8.1

Apply the distributive property.

$10 \int - 1 \sec^{1} (t) + \sec^{2} (t) \sec^{1} (t) d t$

Step 8.2

Simplify each term.

$10 \int - \sec (t) + \sec^{3} (t) d t$

Step 9

Split the single integral into multiple integrals.

$10 (\int - \sec (t) d t + \int \sec^{3} (t) d t)$

Step 10

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

$10 (- \int \sec (t) d t + \int \sec^{3} (t) d t)$

Step 11

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t)$

Step 12

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec (t) \sec^{2} (t) d t)$

Step 13

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan (t) (\sec (t) \tan (t)) d t)$

Step 14

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{1} (t) \tan (t) \sec (t) d t)$

Step 15

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{1} (t) \tan^{1} (t) \sec (t) d t)$

Step 16

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int {\tan (t)}^{1 + 1} \sec (t) d t)$

Step 17

Simplify the expression.

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Step 17.1

Add $1$ and $1$ .

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{2} (t) \sec (t) d t)$

Step 17.2

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) \tan^{2} (t) d t)$

Step 18

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) (- 1 + \sec^{2} (t)) d t)$

Step 19

Simplify by multiplying through.

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Step 19.1

Rewrite the exponentiation as a product.

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) (- 1 + \sec (t) \sec (t)) d t)$

Step 19.2

Apply the distributive property.

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) \cdot - 1 + \sec (t) (\sec (t) \sec (t)) d t)$

Step 19.3

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \cdot \sec (t) + \sec (t) (\sec (t) \sec (t)) d t)$

Step 20

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec (t) \sec (t) d t)$

Step 21

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec^{1} (t) \sec (t) d t)$

Step 22

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{1 + 1} \sec (t) d t)$

Step 23

Add $1$ and $1$ .

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec (t) d t)$

Step 24

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec^{1} (t) d t)$

Step 25

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{2 + 1} d t)$

Step 26

Add $2$ and $1$ .

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{3} (t) d t)$

Step 27

Split the single integral into multiple integrals.

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (\int - 1 \sec (t) d t + \int \sec^{3} (t) d t))$

Step 28

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (- \int \sec (t) d t + \int \sec^{3} (t) d t))$

Step 29

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t))$

Step 30

Simplify by multiplying through.

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Step 30.1

Apply the distributive property.

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 30.2

Multiply $- 1$ by $- 1$ .

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 31

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2} + C)$

Step 32

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

$10 (- (\ln (| \sec (t) + \tan (t) |) + C) + \frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C}{2} + C)$

Step 33

Simplify.

$10 \frac{- \ln (| \sec (t) + \tan (t) |) \cdot 2 + \sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 34

Simplify.

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Step 34.1

Multiply $2$ by $- 1$ .

$10 \frac{- 2 \ln (| \sec (t) + \tan (t) |) + \sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 34.2

Add $- 2 \ln (| \sec (t) + \tan (t) |)$ and $\ln (| \sec (t) + \tan (t) |)$ .

$10 \frac{\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 34.3

Combine $10$ and $\frac{\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)}{2}$ .

$\frac{10 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))}{2} + C$

Step 34.4

Cancel the common factor of $10$ and $2$ .

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Step 34.4.1

Factor $2$ out of $10 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))$ .

$\frac{2 (5 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)))}{2} + C$

Step 34.4.2

Cancel the common factors.

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Step 34.4.2.1

Factor $2$ out of $2$ .

$\frac{2 (5 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)))}{2 (1)} + C$

Step 34.4.2.2

Cancel the common factor.

$\frac{2 (5 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)))}{2 \cdot 1} + C$

Step 34.4.2.3

Rewrite the expression.

$\frac{5 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))}{1} + C$

Step 34.4.2.4

Divide $5 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))$ by $1$ .

$5 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)) + C$

Step 35

Substitute back in for each integration substitution variable.

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Step 35.1

Replace all occurrences of $t$ with $arcsec (\frac{u}{\sqrt{10}})$ .

$5 (\sec (arcsec (\frac{u}{\sqrt{10}})) \tan (arcsec (\frac{u}{\sqrt{10}})) - \ln (| \sec (arcsec (\frac{u}{\sqrt{10}})) + \tan (arcsec (\frac{u}{\sqrt{10}})) |)) + C$

Step 35.2

Replace all occurrences of $u$ with $x + 4$ .

$5 (\sec (arcsec (\frac{x + 4}{\sqrt{10}})) \tan (arcsec (\frac{x + 4}{\sqrt{10}})) - \ln (| \sec (arcsec (\frac{x + 4}{\sqrt{10}})) + \tan (arcsec (\frac{x + 4}{\sqrt{10}})) |)) + C$

Step 36

Reorder terms.

$5 (\sec (arcsec (\frac{1}{\sqrt{10}} (x + 4))) \tan (arcsec (\frac{1}{\sqrt{10}} (x + 4))) - \ln (| \sec (arcsec (\frac{1}{\sqrt{10}} (x + 4))) + \tan (arcsec (\frac{1}{\sqrt{10}} (x + 4))) |)) + C$