Calculus Examples

$\sqrt{x^{2} + 1}$

Step 1

Write $\sqrt{x^{2} + 1}$ as a function.

$f (x) = \sqrt{x^{2} + 1}$

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.

$F (x) = \int \sqrt{x^{2} + 1} d x$

Step 4

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.

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

Step 5

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

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Step 5.1

Apply pythagorean identity.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.2

Add $1$ and $2$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify the expression.

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Step 12.1

Add $1$ and $1$ .

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

Step 12.2

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

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

Step 13

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

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

Step 14

Simplify by multiplying through.

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Step 14.1

Rewrite the exponentiation as a product.

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

Step 14.2

Apply the distributive property.

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

Step 14.3

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

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

Step 15

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

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

Step 16

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

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

Step 17

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

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

Step 18

Add $1$ and $1$ .

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

Step 19

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

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

Step 20

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

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

Step 21

Add $2$ and $1$ .

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

Step 22

Split the single integral into multiple integrals.

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

Step 23

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

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

Step 24

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

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

Step 25

Simplify by multiplying through.

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Step 25.1

Apply the distributive property.

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

Step 25.2

Multiply $- 1$ by $- 1$ .

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

Step 26

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

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

Step 27

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

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

Step 28

Simplify.

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

Step 29

Replace all occurrences of $t$ with $\arctan (x)$ .

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

Step 30

The answer is the antiderivative of the function $f (x) = \sqrt{x^{2} + 1}$ .

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