Question

Use the table of integrals, or a computer or calculator with symbolic integration capabilities, to find the indefinite integral.

∫16/√x^2+16 dx

Answer 1

`\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \ln \big| √(x^2 + 16) + x \big| + C`

General Formulas and Concepts:

Pre-Calculus

• Trigonometric Identities

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Integration

• Integrals
• [Indefinite Integrals] integration Constant C

Integration Property [Multiplied Constant]:
`\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx`

U-Substitution

• Trigonometric Substitution

Explanation:

Step 1: Define

Identify

`\displaystyle \int {(16)/(√(x^2 + 16))} \, dx`

Step 2: Integrate Pt. 1

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \int {(1)/(√(x^2 + 16))} \, dx`
2. [Integrand] Rewrite:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \int {(1)/(√(x^2 + 4^2))} \, dx`

Step 3: Integrate Pt. 2

Set variables for trigonometric substitution.

1. Set trigonometric x:
   `\displaystyle x = 4 \tan (\theta)`
2. [x] Differentiate [Trigonometric Differentiation, Multiplied Constant]:
   `\displaystyle dx = 4 \sec ^2(\theta) \ d\theta`

Step 4: Integrate Pt. 3

1. [Integral] Trigonometric Substitution:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \int {(4 \sec ^2(\theta))/(√([4 \tan (\theta)]^2 + 4^2))} \, d\theta`
2. [Integrand] Expand:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \int {(4 \sec ^2(\theta))/(√(4^2 \tan ^2(\theta) + 4^2))} \, d\theta`
3. [Integrand] Factor:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \int {(4 \sec ^2(\theta))/(√(4^2[ \tan ^2(\theta) + 1]))} \, d\theta`
4. [Integrand] Rewrite [Trigonometric Identity]:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \int {(4 \sec ^2(\theta))/(√(4^2 \sec ^2(\theta)))} \, d\theta`
5. [Integrand] Simplify:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \int {(4 \sec ^2(\theta))/(4 \sec (\theta))} \, d\theta`
6. [Integrand] Simplify:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \int {\sec (\theta)} \, d\theta`
7. [Integral] Trigonometric Integration:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \ln \big| \sec (\theta) + \tan (\theta) \big| + C`
8. [Trig] Substitute [See Attachment]:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \ln \bigg| (√(x^2 + 16))/(4) + (x)/(4) \bigg| + C`
9. Simplify:
   `\displaystyle \int {(16)/(√(x^2 + 16))} \, dx = 16 \ln \big| √(x^2 + 16)} + x \big| + C`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration