Mathematics, 14.01.2020 21:31 madisonbrookecallis

Exercise is mixed —some require integration by parts, while others can be integrated by using techniques discussed in the chapter on integration.
∫^1_0 x^2 dx/2x^3+1

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Answer from: Onlyoneeniyaaa

$\displaystyle \int\limits^1_0 {\frac{x^2}{2x^3 + 1}} \, dx = \frac{\ln 3}{6}$

General Formulas and Concepts:

Calculus

Differentiation

DerivativesDerivative Notation

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

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:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integrals

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]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int\limits^1_0 {\frac{x^2}{2x^3 + 1}} \, dx$

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

Set u: $\displaystyle u = 2x^3 + 1$ [u] Basic Power Rule [Derivative Properties]: $\displaystyle du = 6x^2 \ dx$ [Limits] Swap: $\displaystyle \left \{ {{x = 1 ,\ u = 2(1)^3 + 1 = 3} \atop {x = 0 ,\ u = 2(0)^3 + 1 = 1}} \right.$

Step 3: Integrate Pt. 2

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^1_0 {\frac{x^2}{2x^3 + 1}} \, dx = \frac{1}{6} \int\limits^1_0 {\frac{6x^2}{2x^3 + 1}} \, dx$ [Integral] U-Substitution: $\displaystyle \int\limits^1_0 {\frac{x^2}{2x^3 + 1}} \, dx = \frac{1}{6} \int\limits^3_1 {\frac{1}{u}} \, du$ [Integral] Logarithmic Integration: $\displaystyle \int\limits^1_0 {\frac{x^2}{2x^3 + 1}} \, dx = \frac{1}{6} \ln \big| u \big| \bigg| \limits^3_1$ Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^1_0 {\frac{x^2}{2x^3 + 1}} \, dx = \frac{\ln 3}{6}$

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

Unit: Integration

Answer from: Quest

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step-by-step explanation:

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Answer from: Quest

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answer: d.) the total number of protons and neutrons is 15

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