Mathematics, 05.12.2020 09:30 Ackussinglake63

IF ANYONE HELPS ME WITH THIS I WILL GIVE U BRAINLIEST AND 100 POINTS BTW ITS INTEGRALS FOR CALCULUS

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$\displaystyle \int\limits^6_4 {\frac{1}{x^3}e^{4x^{-2}}} \, dx = \frac{e^\bigg{\frac{1}{4}}}{8} - \frac{e^\bigg{\frac{1}{9}}}{8}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Algebra I

Terms/CoefficientsFactoringExponential Rule [Rewrite]: $\displaystyle b^{-m} = \frac{1}{b^m}$

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integrals

Definite Integrals

Integration Constant C

Integration Rule [Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

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

U-Substitution

eˣ Integration: $\displaystyle \int {e^u} \, dx = e^u + C$

Step-by-step explanation:

Step 1: Define

$\displaystyle \int\limits^6_4 {\frac{1}{x^3}e^{4x^{-2}}} \, dx$

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

Set: $\displaystyle u = 4x^{-2}$ [u] Differentiate [Derivative Rule - Basic Power Rule]: $\displaystyle du = -8x^{-3} \ dx$ [du] Rewrite [Exponential Rule - Rewrite]: $\displaystyle du = \frac{-8}{x^3} \ dx$

Step 3: Integrate Pt. 2

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^6_4 {\frac{1}{x^3}e^{4x^{-2}}} \, dx = \frac{-1}{8}\int\limits^6_4 {\frac{-8}{x^3}e^{4x^{-2}}} \, dx$ [Integral] U-Substitution: $\displaystyle \int\limits^6_4 {\frac{1}{x^3}e^{4x^{-2}}} \, dx = \frac{-1}{8}\int\limits^{\frac{1}{9}}_{\frac{1}{4}} {e^u} \, dx$ [Integral] eˣ Integration: $\displaystyle \int\limits^6_4 {\frac{1}{x^3}e^{4x^{-2}}} \, dx = \frac{-1}{8}(e^u) \bigg| \limits^{\frac{1}{9}}_{\frac{1}{4}}$ Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^6_4 {\frac{1}{x^3}e^{4x^{-2}}} \, dx = \frac{-1}{8} \bigg[ -e^\bigg{\frac{1}{9}} \bigg( e^\bigg{\frac{5}{36}} - 1 \bigg) \bigg]$ Simplify: $\displaystyle \int\limits^6_4 {\frac{1}{x^3}e^{4x^{-2}}} \, dx = \frac{e^\bigg{\frac{1}{4}}}{8} - \frac{e^\bigg{\frac{1}{9}}}{8}$

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

Unit: Integration

Book: College Calculus 10e

Answer from: Quest

Angle onp = 43°angle lno = 137°angle mnp = 137°
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Answer from: Quest

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IF ANYONE HELPS ME WITH THIS I WILL GIVE U BRAINLIEST AND 100 POINTS BTW ITS INTEGRALS FOR CALCULUS...

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