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8(-x*+6x) (x5-15x²+4)3 dx Integrate.

8(-x*+6x) (x5-15x²+4)3 dx Integrate.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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I need help finding out how to do this problem, using the cheat sheet formulas provided
# Calculus Cheat Sheet ## The Basics - **Critical Points:** - \( x \to f(x) = y \) (where \( y \) can be \( \pm, \phi, - \)) - \( x \to f'(x) = \text{slope} \) (slope can be \( \pm, \phi, - \)) - \( \phi \) represents a maximum, minimum, or H.P./R.P. - \( x \to f''(x) = \text{concavity} \) (concavity is \( \cup, \cap, \phi \)) - \( \phi \) indicates a point of inflection. ## Derivatives - **Power Rule:** \( y = x^n \) leads to \( y' = nx^{n-1} \) - **Product Rule:** \( y = uv \), then \( y' = u'v + uv' \) - **Quotient Rule:** \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \) - **Exponential Function:** \( y = e^u \), then \( y' = u'e^u \) - **Logarithmic Function:** \( y = \ln u \), then \( y' = \frac{u'}{u} \) ## Integrals - **Basic Integration:** - \( y = \int x^n dx \rightarrow \frac{x^{n+1}}{n+1} + K, \, n \neq -1 \) - **Integrating Exponential & Logarithmic Functions:** - \( y = \int u'dx \rightarrow u + K \) - \( y = \int e^u du \rightarrow e^u + K \) - \( y = \int \frac{1}{u} du \rightarrow \ln u + K, \, n = -1 \) - **Integration Steps:** 1. Make the integral presentable. 2. Determine \( u \); find \( u' \). 3. Identify what is known and what is needed. 4. Match to known integral templates. 5. Execute the
Transcribed Image Text:# Calculus Cheat Sheet ## The Basics - **Critical Points:** - \( x \to f(x) = y \) (where \( y \) can be \( \pm, \phi, - \)) - \( x \to f'(x) = \text{slope} \) (slope can be \( \pm, \phi, - \)) - \( \phi \) represents a maximum, minimum, or H.P./R.P. - \( x \to f''(x) = \text{concavity} \) (concavity is \( \cup, \cap, \phi \)) - \( \phi \) indicates a point of inflection. ## Derivatives - **Power Rule:** \( y = x^n \) leads to \( y' = nx^{n-1} \) - **Product Rule:** \( y = uv \), then \( y' = u'v + uv' \) - **Quotient Rule:** \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \) - **Exponential Function:** \( y = e^u \), then \( y' = u'e^u \) - **Logarithmic Function:** \( y = \ln u \), then \( y' = \frac{u'}{u} \) ## Integrals - **Basic Integration:** - \( y = \int x^n dx \rightarrow \frac{x^{n+1}}{n+1} + K, \, n \neq -1 \) - **Integrating Exponential & Logarithmic Functions:** - \( y = \int u'dx \rightarrow u + K \) - \( y = \int e^u du \rightarrow e^u + K \) - \( y = \int \frac{1}{u} du \rightarrow \ln u + K, \, n = -1 \) - **Integration Steps:** 1. Make the integral presentable. 2. Determine \( u \); find \( u' \). 3. Identify what is known and what is needed. 4. Match to known integral templates. 5. Execute the
Certainly! Below is the transcription of the image suitable for an educational website: --- **Problem d)** \( y = \int \frac{8(-x^4 + 6x)}{(x^5 - 15x^2 + 4)^3} \, dx \) **Task:** Integrate. --- There are no graphs or diagrams in the image. The problem requires performing the integration of a given rational function. The numerator involves a polynomial expression \( -x^4 + 6x \), and the denominator is a polynomial \( (x^5 - 15x^2 + 4)^3 \) raised to the third power. The integration process will likely involve using techniques suitable for handling rational functions.
Transcribed Image Text:Certainly! Below is the transcription of the image suitable for an educational website: --- **Problem d)** \( y = \int \frac{8(-x^4 + 6x)}{(x^5 - 15x^2 + 4)^3} \, dx \) **Task:** Integrate. --- There are no graphs or diagrams in the image. The problem requires performing the integration of a given rational function. The numerator involves a polynomial expression \( -x^4 + 6x \), and the denominator is a polynomial \( (x^5 - 15x^2 + 4)^3 \) raised to the third power. The integration process will likely involve using techniques suitable for handling rational functions.
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