Question

Evaluate the following Integrals ∫sin

Answer 1

`\displaystyle \int {xsinx} \, dx = -xcosx + sinx + C`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals
• Indefinite Integrals
• Integration Constant C

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

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

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

Explanation:

Step 1: Define

Identify

`\displaystyle \int {xsinx} \, dx`

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

1. Set u:
   `\displaystyle u = x`
2. [u] Differentiate [Basic Power Rule]:
   `\displaystyle du = dx`
3. [dv] Trigonometric Integration:
   `\displaystyle v = -cosx`
4. Set dv:
   `\displaystyle dv = sinx \ dx`

Step 3: Integrate Pt. 2

1. [Integral] Integration by Parts:
   `\displaystyle \int {xsinx} \, dx = -xcosx - \int {-cosx} \, dx`
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {xsinx} \, dx = -xcosx + \int {cosx} \, dx`
3. [Integral] Trigonometric Integration:
   `\displaystyle \int {xsinx} \, dx = -xcosx + sinx + C`

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

Unit: Integration

Book: College Calculus 10e