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Answer:

\(\displaystyle \int {\sin^{-1}(x)} \, dx = x \sin^{-1}(x) + \sqrt{1 - x^2} + C\)

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Addition/Subtraction]:                                                         \(\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]\)

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals
  • [Indefinite Integrals] Integration Constant C

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

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

U-Substitution

Integration by Parts:                                                                                               \(\displaystyle \int {u} \, dv = uv - \int {v} \, du\)

  • [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig

Step-by-step explanation:

Step 1: Define

Identify

\(\displaystyle \int {\sin^{-1}(x)} \, dx\)

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

  1. Set u:                                                                                                             \(\displaystyle u = \sin^{-1}(x)\)
  2. [u] Arctrig Differentiation:                                                                             \(\displaystyle du = \frac{1}{\sqrt{1 - x^2}} \ dx\)
  3. Set dv:                                                                                                           \(\displaystyle dv = dx\)
  4. [dv] Integration Rule [Reverse Power Rule]:                                               \(\displaystyle v = x\)

Step 3: Integrate Pt. 2

  1. [Integral] Integration by Parts:                                                                       \(\displaystyle \int {\sin^{-1}(x)} \, dx = x \sin^{-1}(x) - \int {\frac{x}{\sqrt{1 - x^2}}} \, dx\)

Step 4: Integrate Pt. 3

Identify variables for u-substitution.

  1. Set u:                                                                                                             \(\displaystyle u = 1 - x^2\)
  2. [u] Basic Power Rule [Derivative Property - Addition/Subtraction]:           \(\displaystyle du = -2x \ dx\)

Step 5: Integrate Pt. 4

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 \(\displaystyle \int {\sin^{-1}(x)} \, dx = x \sin^{-1}(x) + \frac{1}{2} \int {\frac{2x}{\sqrt{1 - x^2}}} \, dx\)
  2. [Integral] U-Substitution:                                                                               \(\displaystyle \int {\sin^{-1}(x)} \, dx = x \sin^{-1}(x) + \frac{1}{2} \int {\frac{1}{\sqrt{u}}} \, du\)
  3. [Integral] Integration Rule [Reverse Power Rule]:                                       \(\displaystyle \int {\sin^{-1}(x)} \, dx = x \sin^{-1}(x) + \sqrt{u} + C\)
  4. [u] Back-Substitute:                                                                                       \(\displaystyle \int {\sin^{-1}(x)} \, dx = x \sin^{-1}(x) + \sqrt{1 - x^2} + C\)

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

Unit: Integration

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