Question

What is the resulting integral?

Answer 1

`\displaystyle \int {(2)/((1 + √(x))^5)} \, dx = (-(4√(x) + 1))/(3(√(x) + 1)^4) + C`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Exponential Rule [Root Rewrite]:
  `\displaystyle \sqrt[n]{x} = x^{(1)/(n)}`
• Exponential Rule [Rewrite]:
  `\displaystyle b^(-m) = (1)/(b^m)`

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Antiderivatives - Integrals

Integration Constant C

U-Substitution

• U-Solve

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + C`

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

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

Explanation:

Step 1: Define

`\displaystyle \int {(2)/((1 + √(x))^5)} \, dx`

Step 2: Identify Variables

Find the variables u-solve using u-substitution.

U-Substitution

`\displaystyle u = 1 + √(x)`

`\displaystyle du = (1)/(2√(x))dx`

U-Solve

`\displaystyle x = (u - 1)^2`

`\displaystyle dx = 2√(x)du`

Step 3: Integration

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle 2\int {(1)/((1 + √(x))^5)} \, dx`
2. [Integral] U-Solve:
   `\displaystyle 2\int {(1)/((1 + √((u - 1)^2))^5)} \, 2√(x)du`
3. [Integral] Simplify:
   `\displaystyle 2\int {(2√(x))/(u^5)} \, du`
4. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle 4\int {(√(x))/(u^5)} \, du`
5. [Integral] U-Solve:
   `\displaystyle 4\int {(√((u - 1)^2))/(u^5)} \, du`
6. [Integral] Simplify:
   `\displaystyle 4\int {(u - 1)/(u^5)} \, du`
7. [Integral] Rewrite [Integration Property - Subtraction]:
   `\displaystyle 4[\int {(u)/(u^5)} \, du - \int {(1)/(u^5)} \, du]`
8. [Integrals] Simplify:
   `\displaystyle 4[\int {(1)/(u^4)} \, du - \int {(1)/(u^5)} \, du]`
9. [Integrals] Rewrite [Exponential Rule - Rewrite]:
   `\displaystyle 4[\int {u^(-4)} \, du - \int {u^(-5)} \, du]`
10. [Integrals] Reverse Power Rule:
    `\displaystyle 4[(u^(-3))/(-3) - (u^(-4))/(-4)] + C`
11. Rewrite [Exponential Rule - Rewrite]:
    `\displaystyle 4[(-1)/(3u^3) + (1)/(4u^4)] + C`
12. [Brackets] Multiply:
    `\displaystyle (-4)/(3u^3) + (4)/(4u^4) + C`
13. Back-Substitute:
    `\displaystyle (-4)/(3(1 + √(x))^3) + (4)/(4(1 + √(x))^4) + C`
14. Combine:
    `\displaystyle (-(4√(x) + 1))/(3(√(x) + 1)^4) + C`

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

Unit: Integration

Book: College Calculus 10e