Calculus Examples

\int \sqrt{1 + x^{2}} d x

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Let

x = tan (t)

$x = \tan (t)$ , where

- \frac{π}{2} \leq t \leq \frac{π}{2}

$- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then

d x = {sec}^{2} (t) d t

$d x = \sec^{2} (t) d t$ . Note that since

- \frac{π}{2} \leq t \leq \frac{π}{2}

$- \frac{π}{2} \leq t \leq \frac{π}{2}$ ,

{sec}^{2} (t)

$\sec^{2} (t)$ is positive.

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

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

Step 3

Simplify

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

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

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Step 3.1

Rearrange terms.

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

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

Step 3.2

Apply pythagorean identity.

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

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

Step 3.3

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

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

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

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

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

Step 4

Multiply

sec (t)

$\sec (t)$ by

{sec}^{2} (t)

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

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Step 4.1

Multiply

sec (t)

$\sec (t)$ by

{sec}^{2} (t)

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

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Step 4.1.1

Raise

sec (t)

$\sec (t)$ to the power of

1

$1$ .

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

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

Step 4.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 4.2

Add

1

$1$ and

2

$2$ .

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

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

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

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

Step 5

Factor

sec (t)

$\sec (t)$ out of

{sec}^{3} (t)

$\sec^{3} (t)$ .

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

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

Step 6

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = sec (t)

$u = \sec (t)$ and

d v = {sec}^{2} (t)

$d v = \sec^{2} (t)$ .

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

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

Step 7

Raise

tan (t)

$\tan (t)$ to the power of

1

$1$ .

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

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

Step 8

Raise

tan (t)

$\tan (t)$ to the power of

1

$1$ .

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

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

Step 9

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 10

Simplify the expression.

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Step 10.1

Add

1

$1$ and

1

$1$ .

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

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

Step 10.2

Reorder

{tan}^{2} (t)

$\tan^{2} (t)$ and

sec (t)

$\sec (t)$ .

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

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

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

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

Step 11

Using the Pythagorean Identity, rewrite

{tan}^{2} (t)

$\tan^{2} (t)$ as

- 1 + {sec}^{2} (t)

$- 1 + \sec^{2} (t)$ .

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

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

Step 12

Simplify by multiplying through.

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Step 12.1

Rewrite the exponentiation as a product.

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

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

Step 12.2

Apply the distributive property.

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

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

Step 12.3

Reorder

$\sec (t)$ and

$- 1$ .

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

Step 13

Raise

$\sec (t)$ to the power of

$1$ .

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

Step 14

Raise

$\sec (t)$ to the power of

$1$ .

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

Step 15

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 16

Add

$1$ and

$1$ .

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

Step 17

Raise

$\sec (t)$ to the power of

$1$ .

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

Step 18

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 19

Add

$2$ and

$1$ .

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

Step 20

Split the single integral into multiple integrals.

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

Step 21

Since

$- 1$ is constant with respect to

$t$ , move

$- 1$ out of the integral.

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

Step 22

The integral of

$\sec (t)$ with respect to

$t$ is

$\ln (| \sec (t) + \tan (t) |)$ .

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

Step 23

Simplify by multiplying through.

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Step 23.1

Apply the distributive property.

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

Step 23.2

Multiply

$- 1$ by

$- 1$ .

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

Step 24

Solving for

$\int \sec^{3} (t) d t$ , we find that

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

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

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

Step 25

Multiply

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

$1$ .

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

Step 26

Simplify.

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

Step 27

Replace all occurrences of

$t$ with

$\arctan (x)$ .

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