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

right

step-by-step explanation:

Answer from: Quest

is there answer choices?

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