Question

How to you solve this indefinite integral from 0 to 1? ∫(xsin(πx^2))dx

Answer 1

`\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = (1)/(\pi)`

General Formulas and Concepts:

Pre-Calculus

• Unit Circle

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(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 = (x^(n + 1))/(n + 1) + C`

Integration Rule [Fundamental Theorem of Calculus 1]:
`\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)`

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

U-Substitution

Explanation:

Step 1: Define

Identify

`\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx`

Step 2: Integrate Pt. 1

Set variables for u-substitution.

1. Set u:
   `\displaystyle u = \pi x^2`
2. [u] Differentiate [Basic Power Rule, Multiplied Constant]:
   `\displaystyle du = 2\pi x \ dx`

Step 3: Integrate Pt. 2

1. [Integral] Rewrite:
   `\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = (1)/(2 \pi) \int\limits^1_0 {2\pi xsin(\pi x^2)} \, dx`
2. [Integral] U-Substitution:
   `\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = (1)/(2 \pi) \int\limits^(\pi)_0 {sin(u)} \, du`
3. [Integral] Trigonometric Integration:
   `\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = (1)/(2 \pi)[-cos(u)] \bigg| \limits^(\pi)_0`
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus]:
   `\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = (1)/(2 \pi)(2)`
5. Simplify:
   `\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = (1)/(\pi)`

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

Unit: Integration