Question

$\int \frac{x^{2}-x-8}{(x+1)\left(x^{2}+5 x+6\right)} d x$

   $\int \frac{x^{2}-x-8}{(x+1)\left(x^{2}+5 x+6\right)} d x$

Calculus, Early Transcendentals

Calculus, Early Transcendentals

Michael Sullivan,… 1st Edition

Chapter 7, Problem 28

Instant Answer

Solved by Expert Ma. Theresa Alin

08/07/2021

Step 1

The function is $\frac{x^{2}-x-8}{(x+1)(x^{2}+5x+6)}$. The denominator can be factored as $(x+1)(x+2)(x+3)$. Show more…

Show all steps

lock

Thanks for your feedback!

$\int \frac{x^{2}-x-8}{(x+1)\left(x^{2}+5 x+6\right)} d x$

Play audio

Feedback

Powered by NumerAI

Ma. Theresa Alin and 81 other subject Calculus 2 / BC educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Integration of Rational Functions

This area concerns finding antiderivatives of ratios of polynomials. By applying strategies such as factoring and partial fractions decomposition, the problem of integrating a rational function is reduced to integrating simpler terms, often leading to standard forms such as logarithmic functions for linear denominators.

Partial Fractions Decomposition

This technique is used to rewrite a complex rational function as a sum of simpler fractions whose denominators are the factors of the original denominator. Once decomposed, each simpler fraction is generally easier to integrate individually, making the overall integration process more manageable.

Polynomial Factorization

This concept involves expressing a polynomial as a product of simpler polynomials. In integration problems, particularly those involving rational functions, factoring the denominator (and sometimes the numerator) is a crucial first step that simplifies the integrand, allowing methods like partial fraction decomposition to be applied effectively.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later