Calculus Examples

$5 \sqrt{x^{2} + 6}$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \sqrt{x^{2} + 6} d x$

Step 2

Let $x = \sqrt{6} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \sec^{2} (t) \sqrt{6} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\sec^{2} (t) \sqrt{6}$ is positive.

$5 \int \sqrt{{(\sqrt{6} \tan (t))}^{2} + {\sqrt{6}}^{2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Simplify $\sqrt{{(\sqrt{6} \tan (t))}^{2} + {\sqrt{6}}^{2}}$ .

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

Apply the product rule to $\sqrt{6} \tan (t)$ .

$5 \int \sqrt{{\sqrt{6}}^{2} \tan^{2} (t) + {\sqrt{6}}^{2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 3.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$5 \int \sqrt{{(6^{\frac{1}{2}})}^{2} \tan^{2} (t) + {\sqrt{6}}^{2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.2.2

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

$5 \int \sqrt{6^{\frac{1}{2} \cdot 2} \tan^{2} (t) + {\sqrt{6}}^{2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.2.3

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

$5 \int \sqrt{6^{\frac{2}{2}} \tan^{2} (t) + {\sqrt{6}}^{2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.1.2.4.1

Cancel the common factor.

$5 \int \sqrt{6^{\frac{2}{2}} \tan^{2} (t) + {\sqrt{6}}^{2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.2.4.2

Rewrite the expression.

$5 \int \sqrt{6^{1} \tan^{2} (t) + {\sqrt{6}}^{2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.2.5

Evaluate the exponent.

$5 \int \sqrt{6 \tan^{2} (t) + {\sqrt{6}}^{2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 3.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$5 \int \sqrt{6 \tan^{2} (t) + {(6^{\frac{1}{2}})}^{2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.3.2

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

$5 \int \sqrt{6 \tan^{2} (t) + 6^{\frac{1}{2} \cdot 2}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.3.3

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

$5 \int \sqrt{6 \tan^{2} (t) + 6^{\frac{2}{2}}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.1.3.4.1

Cancel the common factor.

$5 \int \sqrt{6 \tan^{2} (t) + 6^{\frac{2}{2}}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.3.4.2

Rewrite the expression.

$5 \int \sqrt{6 \tan^{2} (t) + 6^{1}} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.1.3.5

Evaluate the exponent.

$5 \int \sqrt{6 \tan^{2} (t) + 6} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.2

Factor $6$ out of $6 \tan^{2} (t) + 6$ .

Tap for more steps...

Step 3.1.2.1

Factor $6$ out of $6 \tan^{2} (t)$ .

$5 \int \sqrt{6 (\tan^{2} (t)) + 6} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.2.2

Factor $6$ out of $6$ .

$5 \int \sqrt{6 (\tan^{2} (t)) + 6 (1)} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.2.3

Factor $6$ out of $6 (\tan^{2} (t)) + 6 (1)$ .

$5 \int \sqrt{6 (\tan^{2} (t) + 1)} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.3

Apply pythagorean identity.

$5 \int \sqrt{6 \sec^{2} (t)} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.4

Reorder $6$ and $\sec^{2} (t)$ .

$5 \int \sqrt{\sec^{2} (t) \cdot 6} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.1.5

Pull terms out from under the radical.

$5 \int \sec (t) \sqrt{6} (\sec^{2} (t) \sqrt{6}) d t$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

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

$5 \int \sec^{2} (t) \sec^{1} (t) \sqrt{6} \sqrt{6} d t$

Step 3.2.2

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

$5 \int {\sec (t)}^{2 + 1} \sqrt{6} \sqrt{6} d t$

Step 3.2.3

Add $2$ and $1$ .

$5 \int \sec^{3} (t) \sqrt{6} \sqrt{6} d t$

Step 3.2.4

Raise $\sqrt{6}$ to the power of $1$ .

$5 \int \sec^{3} (t) ({\sqrt{6}}^{1} \sqrt{6}) d t$

Step 3.2.5

Raise $\sqrt{6}$ to the power of $1$ .

$5 \int \sec^{3} (t) ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) d t$

Step 3.2.6

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

$5 \int \sec^{3} (t) {\sqrt{6}}^{1 + 1} d t$

Step 3.2.7

Add $1$ and $1$ .

$5 \int \sec^{3} (t) {\sqrt{6}}^{2} d t$

Step 3.2.8

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Tap for more steps...

Step 3.2.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$5 \int \sec^{3} (t) {(6^{\frac{1}{2}})}^{2} d t$

Step 3.2.8.2

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

$5 \int \sec^{3} (t) \cdot 6^{\frac{1}{2} \cdot 2} d t$

Step 3.2.8.3

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

$5 \int \sec^{3} (t) \cdot 6^{\frac{2}{2}} d t$

Step 3.2.8.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.8.4.1

Cancel the common factor.

$5 \int \sec^{3} (t) \cdot 6^{\frac{2}{2}} d t$

Step 3.2.8.4.2

Rewrite the expression.

$5 \int \sec^{3} (t) \cdot 6^{1} d t$

Step 3.2.8.5

Evaluate the exponent.

$5 \int \sec^{3} (t) \cdot 6 d t$

Step 3.2.9

Move $6$ to the left of $\sec^{3} (t)$ .

$5 \int 6 \sec^{3} (t) d t$

Step 4

Since $6$ is constant with respect to $t$ , move $6$ out of the integral.

$5 (6 \int \sec^{3} (t) d t)$

Step 5

Simplify with factoring out.

Tap for more steps...

Step 5.1

Multiply $6$ by $5$ .

$30 \int \sec^{3} (t) d t$

Step 5.2

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

$30 \int \sec (t) \sec^{2} (t) d t$

Step 6

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

$30 (\sec (t) \tan (t) - \int \tan (t) (\sec (t) \tan (t)) d t)$

Step 7

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

$30 (\sec (t) \tan (t) - \int \tan^{1} (t) \tan (t) \sec (t) d t)$

Step 8

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

$30 (\sec (t) \tan (t) - \int \tan^{1} (t) \tan^{1} (t) \sec (t) d t)$

Step 9

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

$30 (\sec (t) \tan (t) - \int {\tan (t)}^{1 + 1} \sec (t) d t)$

Step 10

Simplify the expression.

Tap for more steps...

Step 10.1

Add $1$ and $1$ .

$30 (\sec (t) \tan (t) - \int \tan^{2} (t) \sec (t) d t)$

Step 10.2

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

$30 (\sec (t) \tan (t) - \int \sec (t) \tan^{2} (t) d t)$

Step 11

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

$30 (\sec (t) \tan (t) - \int \sec (t) (- 1 + \sec^{2} (t)) d t)$

Step 12

Simplify by multiplying through.

Tap for more steps...

Step 12.1

Rewrite the exponentiation as a product.

$30 (\sec (t) \tan (t) - \int \sec (t) (- 1 + \sec (t) \sec (t)) d t)$

Step 12.2

Apply the distributive property.

$30 (\sec (t) \tan (t) - \int \sec (t) \cdot - 1 + \sec (t) (\sec (t) \sec (t)) d t)$

Step 12.3

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

$30 (\sec (t) \tan (t) - \int - 1 \cdot \sec (t) + \sec (t) (\sec (t) \sec (t)) d t)$

Step 13

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

$30 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec (t) \sec (t) d t)$

Step 14

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

$30 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec^{1} (t) \sec (t) d t)$

Step 15

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

$30 (\sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{1 + 1} \sec (t) d t)$

Step 16

Add $1$ and $1$ .

$30 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec (t) d t)$

Step 17

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

$30 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec^{1} (t) d t)$

Step 18

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

$30 (\sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{2 + 1} d t)$

Step 19

Add $2$ and $1$ .

$30 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{3} (t) d t)$

Step 20

Split the single integral into multiple integrals.

$30 (\sec (t) \tan (t) - (\int - 1 \sec (t) d t + \int \sec^{3} (t) d t))$

Step 21

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

$30 (\sec (t) \tan (t) - (- \int \sec (t) d t + \int \sec^{3} (t) d t))$

Step 22

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

$30 (\sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t))$

Step 23

Simplify by multiplying through.

Tap for more steps...

Step 23.1

Apply the distributive property.

$30 (\sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 23.2

Multiply $- 1$ by $- 1$ .

$30 (\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 24

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

$30 (\frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2} + C)$

Step 25

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

$30 (\frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C}{2} + C)$

Step 26

Simplify.

$30 (\frac{1}{2}) (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 27

Simplify.

Tap for more steps...

Step 27.1

Combine $30$ and $\frac{1}{2}$ .

$\frac{30}{2} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 27.2

Cancel the common factor of $30$ and $2$ .

Tap for more steps...

Step 27.2.1

Factor $2$ out of $30$ .

$\frac{2 \cdot 15}{2} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 27.2.2

Cancel the common factors.

Tap for more steps...

Step 27.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 15}{2 (1)} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 27.2.2.2

Cancel the common factor.

$\frac{2 \cdot 15}{2 \cdot 1} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 27.2.2.3

Rewrite the expression.

$\frac{15}{1} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 27.2.2.4

Divide $15$ by $1$ .

$15 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 28

Replace all occurrences of $t$ with $\arctan (\frac{x}{\sqrt{6}})$ .

$15 (\sec (\arctan (\frac{x}{\sqrt{6}})) \tan (\arctan (\frac{x}{\sqrt{6}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{6}})) + \tan (\arctan (\frac{x}{\sqrt{6}})) |)) + C$

Step 29

Reorder terms.

$15 (\sec (\arctan (\frac{1}{\sqrt{6}} x)) \tan (\arctan (\frac{1}{\sqrt{6}} x)) + \ln (| \sec (\arctan (\frac{1}{\sqrt{6}} x)) + \tan (\arctan (\frac{1}{\sqrt{6}} x)) |)) + C$