Mathematics, 27.12.2019 03:31, ajay67

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

Answers: 2

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Answers

answered: kedjenpierrelouis

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

General Formulas and Concepts:

Pre-Calculus

Unit Circle

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Multiplied Constant]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Basic Power Rule:

f(x) = cxⁿf’(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 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

Step-by-step 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.

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

Step 3: Integrate Pt. 2

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

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

Unit: Integration

answered: Guest

step-by-step explanation:

answered: Guest

1e+27

step-by-step explanation:

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