Mathematics, 28.12.2020 14:00 raquelqueengucci25

Integrate each of the following 12 - lnx

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Answer from: joejonaslover7476

$\displaystyle \int {(12 - lnx)} \, dx = x[13 - ln(x)] + C$

General Formulas and Concepts:

Algebra I

Terms/CoefficientsFactoring

Calculus

Differentiation

DerivativesDerivative Notation

Integration

IntegralsIndefinite IntegralsIntegration Constant C

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Integration Property [Addition/Subtraction]: $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

U-Substitution

U-Solve

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

$\displaystyle \int {(12 - lnx)} \, dx$

Step 2: Integrate Pt. 1

[Integral] Rewrite [Integration Property - Addition/Subtraction]: $\displaystyle \int {(12 - lnx)} \, dx = \int {12} \, dx - \int {lnx} \, dx$ [1st Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {(12 - lnx)} \, dx = 12\int {} \, dx - \int {lnx} \, dx$ [1st Integral] Reverse Power Rule: $\displaystyle \int {(12 - lnx)} \, dx = 12x - \int {lnx} \, dx$

Step 3: Integrate Pt. 2

Identify variables for integration by parts using LIPET.

Set u: $\displaystyle u = lnx$ [u] Differentiate [Logarithmic Differentiation]: $\displaystyle du = \frac{1}{x} \ dx$ Set dv: $\displaystyle dv = dx$ [dv] Integration Rule [Reverse Power Rule]: $\displaystyle v = x$

Step 4: Integrate Pt. 3

[Integral] Integration by Parts: $\displaystyle \int {(12 - lnx)} \, dx = 12x - \bigg[ xlnx - \int { \bigg( x \cdot \frac{1}{x} \bigg) } \, dx \bigg]$ [Integrand] Simplify: $\displaystyle \int {(12 - lnx)} \, dx = 12x - \bigg[ xlnx - \int {} \, dx \bigg]$ [Integral] Reverse Power Rule: $\displaystyle \int {(12 - lnx)} \, dx = 12x - \bigg[ xlnx - x + C \bigg]$ Simplify: $\displaystyle \int {(12 - lnx)} \, dx = 12x - xlnx + x + C$ Factor: $\displaystyle \int {(12 - lnx)} \, dx = x[12 - ln(x) + 1] + C$ Simplify: $\displaystyle \int {(12 - lnx)} \, dx = x[13 - ln(x)] + C$

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

Unit: Integration

Book: College Calculus 10e

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Integrate each of the following 12 - lnx...