Calculus Examples

$\sqrt{1 + 4 x^{2}}$

Step 1

Write $\sqrt{1 + 4 x^{2}}$ as a function.

$f (x) = \sqrt{1 + 4 x^{2}}$

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.

$F (x) = \int \sqrt{1 + 4 x^{2}} d x$

Step 4

Let $x = \frac{1}{2} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sec^{2} (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sec^{2} (t)}{2}$ is positive.

$\int \sqrt{1 + 4 {(\frac{1}{2} \tan (t))}^{2}} \frac{\sec^{2} (t)}{2} d t$

Step 5

Simplify terms.

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Step 5.1

Simplify $\sqrt{1 + 4 {(\frac{1}{2} \tan (t))}^{2}}$ .

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Combine $\frac{1}{2}$ and $\tan (t)$ .

$\int \sqrt{1 + 4 {(\frac{\tan (t)}{2})}^{2}} \frac{\sec^{2} (t)}{2} d t$

Step 5.1.1.2

Apply the product rule to $\frac{\tan (t)}{2}$ .

$\int \sqrt{1 + 4 \frac{\tan^{2} (t)}{2^{2}}} \frac{\sec^{2} (t)}{2} d t$

Step 5.1.1.3

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

$\int \sqrt{1 + 4 \frac{\tan^{2} (t)}{4}} \frac{\sec^{2} (t)}{2} d t$

Step 5.1.1.4

Cancel the common factor of $4$ .

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Step 5.1.1.4.1

Cancel the common factor.

$\int \sqrt{1 + 4 \frac{\tan^{2} (t)}{4}} \frac{\sec^{2} (t)}{2} d t$

Step 5.1.1.4.2

Rewrite the expression.

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

Step 5.1.2

Rearrange terms.

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

Step 5.1.3

Apply pythagorean identity.

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

Step 5.1.4

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

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

Step 5.2

Simplify.

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Step 5.2.1

Combine $\sec (t)$ and $\frac{\sec^{2} (t)}{2}$ .

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

Step 5.2.2

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

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Step 5.2.2.1

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

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Step 5.2.2.1.1

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

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

Step 5.2.2.1.2

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

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

Step 5.2.2.2

Add $1$ and $2$ .

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

Step 6

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify the expression.

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Step 12.1

Add $1$ and $1$ .

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

Step 12.2

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

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

Step 13

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

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

Step 14

Simplify by multiplying through.

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Step 14.1

Rewrite the exponentiation as a product.

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

Step 14.2

Apply the distributive property.

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

Step 14.3

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

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

Step 15

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

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

Step 16

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

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

Step 17

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

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

Step 18

Add $1$ and $1$ .

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

Step 19

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

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

Step 20

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

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

Step 21

Add $2$ and $1$ .

$\frac{1}{2} (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{3} (t) d t)$

Step 22

Split the single integral into multiple integrals.

$\frac{1}{2} (\sec (t) \tan (t) - (\int - 1 \sec (t) d t + \int \sec^{3} (t) d t))$

Step 23

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

$\frac{1}{2} (\sec (t) \tan (t) - (- \int \sec (t) d t + \int \sec^{3} (t) d t))$

Step 24

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

$\frac{1}{2} (\sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t))$

Step 25

Simplify by multiplying through.

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Step 25.1

Apply the distributive property.

$\frac{1}{2} (\sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 25.2

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} (\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 26

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

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

Step 27

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

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

Step 28

Simplify.

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

Step 29

Simplify.

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Step 29.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 29.2

Multiply $2$ by $2$ .

$\frac{1}{4} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 30

Replace all occurrences of $t$ with $\arctan (2 x)$ .

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

Step 31

Simplify.

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Step 31.1

Simplify each term.

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Step 31.1.1

Draw a triangle in the plane with vertices $(1, 2 x)$ , $(1, 0)$ , and the origin. Then $\arctan (2 x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, 2 x)$ . Therefore, $\sec (\arctan (2 x))$ is $\sqrt{1 + {(2 x)}^{2}}$ .

$\frac{1}{4} (\sqrt{1 + {(2 x)}^{2}} \tan (\arctan (2 x)) + \ln (| \sec (\arctan (2 x)) + \tan (\arctan (2 x)) |)) + C$

Step 31.1.2

Apply the product rule to $2 x$ .

$\frac{1}{4} (\sqrt{1 + 2^{2} x^{2}} \tan (\arctan (2 x)) + \ln (| \sec (\arctan (2 x)) + \tan (\arctan (2 x)) |)) + C$

Step 31.1.3

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

$\frac{1}{4} (\sqrt{1 + 4 x^{2}} \tan (\arctan (2 x)) + \ln (| \sec (\arctan (2 x)) + \tan (\arctan (2 x)) |)) + C$

Step 31.1.4

The functions tangent and arctangent are inverses.

$\frac{1}{4} (\sqrt{1 + 4 x^{2}} (2 x) + \ln (| \sec (\arctan (2 x)) + \tan (\arctan (2 x)) |)) + C$

Step 31.1.5

Rewrite using the commutative property of multiplication.

$\frac{1}{4} (2 \sqrt{1 + 4 x^{2}} x + \ln (| \sec (\arctan (2 x)) + \tan (\arctan (2 x)) |)) + C$

Step 31.1.6

Simplify each term.

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Step 31.1.6.1

$\frac{1}{4} (2 \sqrt{1 + 4 x^{2}} x + \ln (| \sqrt{1 + {(2 x)}^{2}} + \tan (\arctan (2 x)) |)) + C$

Step 31.1.6.2

Apply the product rule to $2 x$ .

$\frac{1}{4} (2 \sqrt{1 + 4 x^{2}} x + \ln (| \sqrt{1 + 2^{2} x^{2}} + \tan (\arctan (2 x)) |)) + C$

Step 31.1.6.3

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

$\frac{1}{4} (2 \sqrt{1 + 4 x^{2}} x + \ln (| \sqrt{1 + 4 x^{2}} + \tan (\arctan (2 x)) |)) + C$

Step 31.1.6.4

The functions tangent and arctangent are inverses.

$\frac{1}{4} (2 \sqrt{1 + 4 x^{2}} x + \ln (| \sqrt{1 + 4 x^{2}} + 2 x |)) + C$

Step 31.2

Apply the distributive property.

$\frac{1}{4} (2 \sqrt{1 + 4 x^{2}} x) + \frac{1}{4} \ln (| \sqrt{1 + 4 x^{2}} + 2 x |) + C$

Step 31.3

Cancel the common factor of $2$ .

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Step 31.3.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (2 \sqrt{1 + 4 x^{2}} x) + \frac{1}{4} \ln (| \sqrt{1 + 4 x^{2}} + 2 x |) + C$

Step 31.3.2

Factor $2$ out of $2 \sqrt{1 + 4 x^{2}} x$ .

$\frac{1}{2 (2)} (2 (\sqrt{1 + 4 x^{2}} x)) + \frac{1}{4} \ln (| \sqrt{1 + 4 x^{2}} + 2 x |) + C$

Step 31.3.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 (\sqrt{1 + 4 x^{2}} x)) + \frac{1}{4} \ln (| \sqrt{1 + 4 x^{2}} + 2 x |) + C$

Step 31.3.4

Rewrite the expression.

$\frac{1}{2} (\sqrt{1 + 4 x^{2}} x) + \frac{1}{4} \ln (| \sqrt{1 + 4 x^{2}} + 2 x |) + C$

Step 31.4

Combine $\sqrt{1 + 4 x^{2}}$ and $\frac{1}{2}$ .

$\frac{\sqrt{1 + 4 x^{2}}}{2} x + \frac{1}{4} \ln (| \sqrt{1 + 4 x^{2}} + 2 x |) + C$

Step 31.5

Combine $\frac{\sqrt{1 + 4 x^{2}}}{2}$ and $x$ .

$\frac{\sqrt{1 + 4 x^{2}} x}{2} + \frac{1}{4} \ln (| \sqrt{1 + 4 x^{2}} + 2 x |) + C$

Step 31.6

Combine $\frac{1}{4}$ and $\ln (| \sqrt{1 + 4 x^{2}} + 2 x |)$ .

$\frac{\sqrt{1 + 4 x^{2}} x}{2} + \frac{\ln (| \sqrt{1 + 4 x^{2}} + 2 x |)}{4} + C$

Step 31.7

To write $\frac{\sqrt{1 + 4 x^{2}} x}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt{1 + 4 x^{2}} x}{2} \cdot \frac{2}{2} + \frac{\ln (| \sqrt{1 + 4 x^{2}} + 2 x |)}{4} + C$

Step 31.8

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 31.8.1

Multiply $\frac{\sqrt{1 + 4 x^{2}} x}{2}$ by $\frac{2}{2}$ .

$\frac{\sqrt{1 + 4 x^{2}} x \cdot 2}{2 \cdot 2} + \frac{\ln (| \sqrt{1 + 4 x^{2}} + 2 x |)}{4} + C$

Step 31.8.2

Multiply $2$ by $2$ .

$\frac{\sqrt{1 + 4 x^{2}} x \cdot 2}{4} + \frac{\ln (| \sqrt{1 + 4 x^{2}} + 2 x |)}{4} + C$

Step 31.9

Combine the numerators over the common denominator.

$\frac{\sqrt{1 + 4 x^{2}} x \cdot 2 + \ln (| \sqrt{1 + 4 x^{2}} + 2 x |)}{4} + C$

Step 31.10

Move $2$ to the left of $\sqrt{1 + 4 x^{2}} x$ .

$\frac{2 \sqrt{1 + 4 x^{2}} x + \ln (| \sqrt{1 + 4 x^{2}} + 2 x |)}{4} + C$

Step 32

Reorder terms.

$\frac{1}{4} (2 \sqrt{1 + 4 x^{2}} x + \ln (| \sqrt{1 + 4 x^{2}} + 2 x |)) + C$

Step 33

The answer is the antiderivative of the function $f (x) = \sqrt{1 + 4 x^{2}}$ .

$F (x) =$ $\frac{1}{4} (2 \sqrt{1 + 4 x^{2}} x + \ln (| \sqrt{1 + 4 x^{2}} + 2 x |)) + C$