Question

30pts! Brainliest! Calculus 2!
Definite integral of (e^(1/x))/(4x^2) * dx

Please show clear WORK and STEPS for award! Thank you so much!

Answer 1

Answer:

$\displaystyle \int {\frac{e^\big{\frac{1}{x}}}{4x^2}} \, dx = \frac{-e^\big{\frac{1}{x}}}{4} + C$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:                                                                           $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Integration

• Integrals
• [Indefinite Integrals] Integration Constant C

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

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int {\frac{e^\big{\frac{1}{x}}}{4x^2}} \, dx$

Step 2: Integrate Pt. 1

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 $\displaystyle \int {\frac{e^\big{\frac{1}{x}}}{4x^2}} \, dx = \frac{1}{4}\int {\frac{e^\big{\frac{1}{x}}}{x^2}} \, dx$

Step 3: Integrate Pt. 2

Identify variables for u-substitution.

1. Set u:                                                                                                             $\displaystyle u = \frac{1}{x}$
2. [u] Differentiate [Derivative Rules, Basic Power Rule]:                               $\displaystyle du = \frac{-1}{x^2} \ dx$

Step 4: Integrate Pt. 3

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

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

Unit: Integration