Mathematics, 14.01.2020 20:31 dummynate

Use integration by parts to find the integrals in exercise.
∫(4x-12)e-8x dx.

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Answer from: anyawebb06

$\displaystyle \int {(4x - 12)e^{-8x}} \, dx = \frac{e^{-8x}}{2} \bigg( \frac{23}{8} - x \bigg) + C$

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[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:

Step 1: Define

Identify

$\displaystyle \int {(4x - 12)e^{-8x}} \, dx$

Step 2: Integrate Pt. 1

[Integrand] Rewrite [Factor]: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = \int {4(x - 3)e^{-8x}} \, dx$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = 4 \int {(x - 3)e^{-8x}} \, dx$

Step 3: Integrate Pt. 2

Identify variables for integration by parts using LIPET.

Set u: $\displaystyle u = x - 3$ [u] Basic Power Rule [Derivative Property - Addition/Subtraction]: $\displaystyle du = dx$ Set dv: $\displaystyle dv = e^{-8x} \ dx$ [dv] Exponential Integration [U-Substitution]: $\displaystyle v = \frac{-e^{-8x}}{8}$

Step 4: Integrate Pt. 3

[Integral] Integration by Parts: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = 4 \bigg( \frac{-(x - 3)e^{-8x}}{8} - \int {\frac{-e^{-8x}}{8}} \, dx \bigg)$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = 4 \bigg( \frac{-(x - 3)e^{-8x}}{8} + \frac{1}{8} \int {e^{-8x}} \, dx \bigg)$ Factor: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = \frac{1}{2} \bigg( -(x - 3)e^{-8x} + \int {e^{-8x}} \, dx \bigg)$

Step 5: Integrate Pt. 4

Identify variables for u-substitution.

Set u: $\displaystyle u = -8x$ [u] Basic Power Rule [Derivative Rule - Multiplied Constant]: $\displaystyle du = -8 \ dx$

Step 6: Integrate Pt. 5

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = \frac{1}{2} \bigg( -(x - 3)e^{-8x} - \frac{1}{8} \int {-8e^{-8x}} \, dx \bigg)$ [Integral] U-Substitution: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = \frac{1}{2} \bigg( (x - 3)e^{-8x} - \frac{1}{8} \int {e^u} \, dx \bigg)$ [Integral] Exponential Integration: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = \frac{1}{2} \bigg( (x - 3)e^{-8x} - \frac{e^u}{8} \bigg) + C$ [u] Back-Substitute: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = \frac{1}{2} \bigg( (x - 3)e^{-8x} - \frac{e^{-8x}}{8} \bigg) + C$ Factor: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = \frac{e^{-8x}}{2} \bigg( -(x - 3) - \frac{1}{8} \bigg) + C$ Simplify: $\displaystyle \int {(4x - 12)e^{-8x}} \, dx = \frac{e^{-8x}}{2} \bigg( \frac{23}{8} - x \bigg) + C$

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

Unit: Integration

Answer from: Quest

The correct answer is c

Answer from: Quest

A. dividing 99 by 204 will tell you that 99 out of 204 of the people were females at the action movie.

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Use integration by parts to find the integrals in exercise.
∫(4x-12)e-8x dx....

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