Question

Evaluate the integral of xe^(-5x) dx using integrate by parts.

Answer 1

Answer:

$\displaystyle \int {xe^{-5x}} \, dx = -e^{-5x} \bigg( \frac{x}{5} + \frac{1}{25} \bigg) + C$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(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

$\displaystyle \int {xe^{-5x}} \, dx$

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

1. Set u:                                                                                                             $\displaystyle u = x$
2. [u] Differentiate [Derivative Rule - Basic Power Rule]:                               $\displaystyle du = dx$
3. Set dv:                                                                                                           $\displaystyle dv = e^{-5x} \ dx$
4. [dv] Integrate [Exponential Integration, U-Substitution]:                             $\displaystyle v = \frac{-e^{-5x}}{5}$

Step 3: Integrate Pt. 2

1. [Integral] Integration by parts:                                                                       $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \int {\frac{-e^{-5x}}{5}} \, dx$
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} + \frac{1}{5} \int {e^{-5x}} \, dx$

Step 4: Integrate Pt. 3

Identify variables for u-substitution.

1. Set u:                                                                                                             $\displaystyle u = -5x$
2. [u] Differentiate [Derivative Property, Basic Power Rule]:                        $\displaystyle du = -5 \ dx$

Step 5: Integrate Pt. 4

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \frac{1}{25} \int {-5e^{-5x}} \, dx$
2. [Integral] U-Substitution:                                                                               $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \frac{1}{25} \int {e^u} \, du$
3. [Integral] Exponential Integration:                                                               $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \frac{e^u}{25} + C$
4. [u] Back-Substitute:                                                                                       $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \frac{e^{-5x}}{25} + C$
5. Factor:                                                                                                           $\displaystyle \int {xe^{-5x}} \, dx = -e^{-5x} \bigg( \frac{x}{5} + \frac{1}{25} \bigg) + C$

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

Unit: Integration