Mathematics, 06.02.2021 04:00 ugum

Evaluate the integral using integration by parts with the indicated choices of u and dv. (Use C for the constant of integration.) xe7x dx; u

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Answer from: mtiger

$\displaystyle \int {xe^{7x}} \, dx = \frac{e^{7x}}{7} \bigg( x - \frac{1}{7} \bigg) + C$

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

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

$\displaystyle \int {xe^{7x}} \, dx$

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

Set u: $\displaystyle u = x$ [u] Basic Power Rule: $\displaystyle du = dx$ Set dv: $\displaystyle dv = e^{7x} \ dx$ [dv] Exponential Integration [U-Substitution]: $\displaystyle v = \frac{e^{7x}}{7}$

Step 3: integrate Pt. 2

[Integral] Integration by Parts: $\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \int {\frac{e^{7x}}{7}} \, dx$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{1}{7} \int {e^{7x}} \, dx$

Step 4: Integrate Pt. 3

Identify variables for u-substitution.

Set u: $\displaystyle u = 7x$ [u] Basic Power Rule [Derivative Property - Multiplied Constant]: $\displaystyle du = 7 \ dx$

Step 5: Integrate Pt. 4

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{1}{49} \int {7e^{7x}} \, dx$ [Integral] U-Substitution: $\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{1}{49} \int {e^u} \, dx$ [Integral] Exponential Integration: $\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{e^u}{49} + C$ [u] Back-Substitute: $\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{e^{7x}}{49} + C$ Factor: $\displaystyle \int {xe^{7x}} \, dx = \frac{e^{7x}}{7} \bigg( x - \frac{1}{7} \bigg) + C$

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

Unit: Integration

Answer from: Quest

i'm not sure i quite understand, could you provide more information?

Answer from: Quest

C. expression because any expression or equation may contain more than one variable

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Evaluate the integral using integration by parts with the indicated choices of u and dv. (Use C for...