Mathematics, 14.01.2020 21:31 ellieschiller0pdz93k

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

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Answer from: misslux

$\displaystyle \int {x^2e^{2x}} \, dx = \frac{e^{2x}}{2} \bigg( x^2 - x + \frac{1}{2} \bigg) + C$

General Formulas and Concepts:

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 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:

*Note:

You can use tabular integration instead of integrating by parts twice.

Step 1: Define

Identify

$\displaystyle \int {x^2e^{2x}} \, dx$

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

Set u: $\displaystyle u = x^2$ [u] Basic Power Rule: $\displaystyle du = 2x \ dx$ Set dv: $\displaystyle dv = e^{2x} \ dx$ [dv] Exponential Integration [U-Substitution]: $\displaystyle v = \frac{e^{2x}}{2}$

Step 3: Integrate Pt. 2

[Integral] Integration by Parts: $\displaystyle \int {x^2e^{2x}} \, dx = \frac{x^2e^{2x}}{2} - \int {xe^{2x}} \, dx$

Step 4: Integrate Pt. 3

Identify variables for integration by parts using LIPET.

Set u: $\displaystyle u = x$ [u] Basic Power Rule: $\displaystyle du = dx$ Set dv: $\displaystyle dv = e^{2x} \ dx$ [dv] Exponential Integration [U-Substitution]: $\displaystyle v = \frac{e^{2x}}{2}$

Step 5: Integrate Pt. 4

[Integral] Integration by Parts: $\displaystyle \int {x^2e^{2x}} \, dx = \frac{x^2e^{2x}}{2} - \bigg( \frac{xe^{2x}}{2} - \int {\frac{e^{2x}}{2}} \, dx \bigg)$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {x^2e^{2x}} \, dx = \frac{x^2e^{2x}}{2} - \bigg( \frac{xe^{2x}}{2} - \frac{1}{2} \int {e^{2x}} \, dx \bigg)$

Step 6: Integrate Pt. 5

Identify variables for u-substitution.

Set u: $\displaystyle u = 2x$ [u] Basic Power Rule [Derivative Property - Multiplied Constant]: $\displaystyle du = 2 \ dx$

Step 7: Integrate Pt. 6

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {x^2e^{2x}} \, dx = \frac{x^2e^{2x}}{2} - \bigg( \frac{xe^{2x}}{2} - \frac{1}{4} \int {2e^{2x}} \, dx \bigg)$ [Integral] U-Substitution: $\displaystyle \int {x^2e^{2x}} \, dx = \frac{x^2e^{2x}}{2} - \bigg( \frac{xe^{2x}}{2} - \frac{1}{4} \int {e^{u}} \, dx \bigg)$ [Integral] Exponential Integration: $\displaystyle \int {x^2e^{2x}} \, dx = \frac{x^2e^{2x}}{2} - \bigg( \frac{xe^{2x}}{2} - \frac{e^{u}}{4} \bigg) + C$ [u] Back-Substitute: $\displaystyle \int {x^2e^{2x}} \, dx = \frac{x^2e^{2x}}{2} - \bigg( \frac{xe^{2x}}{2} - \frac{e^{2x}}{4} \bigg) + C$ Expand: $\displaystyle \int {x^2e^{2x}} \, dx = \frac{x^2e^{2x}}{2} - \frac{xe^{2x}}{2} + \frac{e^{2x}}{4} + C$ Factor: $\displaystyle \int {x^2e^{2x}} \, dx = \frac{e^{2x}}{2} \bigg( x^2 - x + \frac{1}{2} \bigg) + C$

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

Unit: Integration

Answer from: Quest

pi x^2

step-by-step explanation:

pi times x squared

Answer from: Quest

there are 70 more legs than eyes at the dog show (196 – 126 = 70). dogs have two more legs than eyes, and humans have the same number of legs as eyes. thus 70 is twice the number of dogs at the show, so the number of dogs is half of 70, which is 35. to get the number of humans, subtract 35 from 63: 63 – 35 = 28.

step-by-step explanation:

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Exercise is mixed —some require integration by parts, while others can be integrated by using techni...