Calculus Examples

$\frac{d y}{d x} = \sqrt{1 + x^{2}}$

Step 1

Rewrite the equation.

$d y = \sqrt{1 + x^{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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.

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

Step 2.3.2

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

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Step 2.3.2.1

Rearrange terms.

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

Step 2.3.2.2

Apply pythagorean identity.

$y + C_{1} = \int \sqrt{\sec^{2} (t)} \sec^{2} (t) d t$

Step 2.3.2.3

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

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

Step 2.3.3

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

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Step 2.3.3.1

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

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Step 2.3.3.1.1

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

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

Step 2.3.3.1.2

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

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

Step 2.3.3.2

Add $1$ and $2$ .

$y + C_{1} = \int \sec^{3} (t) d t$

Step 2.3.4

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

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

Step 2.3.5

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

$y + C_{1} = \sec (t) \tan (t) - \int \tan (t) (\sec (t) \tan (t)) d t$

Step 2.3.6

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

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

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

Simplify the expression.

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Step 2.3.9.1

Add $1$ and $1$ .

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

Step 2.3.9.2

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

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

Step 2.3.10

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

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

Step 2.3.11

Simplify by multiplying through.

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Step 2.3.11.1

Rewrite the exponentiation as a product.

$y + C_{1} = \sec (t) \tan (t) - \int \sec (t) (- 1 + \sec (t) \sec (t)) d t$

Step 2.3.11.2

Apply the distributive property.

$y + C_{1} = \sec (t) \tan (t) - \int \sec (t) \cdot - 1 + \sec (t) (\sec (t) \sec (t)) d t$

Step 2.3.11.3

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

$y + C_{1} = \sec (t) \tan (t) - \int - 1 \cdot \sec (t) + \sec (t) (\sec (t) \sec (t)) d t$

Step 2.3.12

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

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

Step 2.3.13

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

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

Step 2.3.14

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

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

Step 2.3.15

Add $1$ and $1$ .

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

Step 2.3.16

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

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

Step 2.3.17

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

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

Step 2.3.18

Add $2$ and $1$ .

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

Step 2.3.19

Split the single integral into multiple integrals.

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

Step 2.3.20

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

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

Step 2.3.21

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

$y + C_{1} = \sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C_{2}) + \int \sec^{3} (t) d t)$

Step 2.3.22

Simplify by multiplying through.

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Step 2.3.22.1

Apply the distributive property.

$y + C_{1} = \sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C_{2}) - \int \sec^{3} (t) d t$

Step 2.3.22.2

Multiply $- 1$ by $- 1$ .

$y + C_{1} = \sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C_{2}) - \int \sec^{3} (t) d t$

Step 2.3.23

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

$y + C_{1} = \frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C_{2})}{2} + C_{3}$

Step 2.3.24

Multiply $\ln (| \sec (t) + \tan (t) |) + C_{2}$ by $1$ .

$y + C_{1} = \frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C_{2}}{2} + C_{3}$

Step 2.3.25

Simplify.

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

Step 2.3.26

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

$y + C_{1} = \frac{1}{2} (\sec (\arctan (x)) \tan (\arctan (x)) + \ln (| \sec (\arctan (x)) + \tan (\arctan (x)) |)) + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

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