7-38 Evaluate the integral. ∫(3 x^2+x+4)/(x^4+3 x^2+2) d x |...

Name: Problem 30, Chapter 7: Calculus Early Transcendentals
Uploaded: 2020-10-24T16:34:41-08:00
Duration: 5 min 58 s
Channel: Audrey Fong
Description: 7-38 Evaluate the integral. ∫(3 x^2+x+4)/(x^4+3 x^2+2) d x

Key Concepts

Integration of Rational Functions

Integrating rational functions often requires rewriting them in a form that can be integrated term-by-term. After applying methods such as partial fraction decomposition, each term typically corresponds to a standard integral form, making the evaluation process more straightforward.

Standard Integral Formulas

These formulas include known integrals such as the arctan function for integrals of the form ?dx/(x^2+a^2) and logarithmic functions for integrals involving rational functions. Recognizing these forms is key to efficiently computing the definite or indefinite integrals of decomposed fractions.

Partial Fraction Decomposition

This is a technique used to express a complex rational function as a sum of simpler fractions whose integrals are known or more easily computed. It is especially useful when the denominator factors into lower-degree polynomials, as it allows the integral to be broken down into manageable parts.

Polynomial Factorization

This concept involves rewriting a polynomial as a product of simpler polynomials. In the context of evaluating integrals, factorization of the denominator simplifies the rational function and prepares it for methods like partial fraction decomposition.

Transcript

00:01 We want to integrate this function, but first let's split it into its partial fraction.

00:06 So i'm going to let the function.

00:12 Notice that the denominator can be factorized into x square plus 1, x squared plus 2.

00:21 So i'm going to let it be a x plus b over x squared plus 1, plus cx plus d over x square plus 2.

00:34 Crossing up the denominator to the right side, you will get this.

00:53 Okay.

00:54 Now because there are four unknowns, so there will be four equations that you need to do, you can let x be anything, so let x be zero, sub into the equation here.

01:08 You will get four equals to 2b plus d.

01:14 When you let x equals to 1, sub it in and rearrange, you will get this.

01:26 So that's the second equation.

01:28 You let x equals to 1.

01:29 You let x equals to 1, sub it in and rearrange x b minus 1 you will get this so that's the third equation and when you let x b 2 you will get this so that's the fourth equation plug that into a calculator that can solve this you get a is 1 b is 1 c is minus 1 d is 2 so now we're ready to speed our function into its partial fraction and that will be 1 plus x over x squared plus 1 plus minus x plus 2 over x squared plus 2.

02:35 So for the first term here, i'm going to split it into x over x square plus 1.

02:46 And then this one here will be plus 1 over x square plus 1.

02:51 For this term here, it will be minus x over x square plus 2.

02:59 And i will have plus 2 over x square plus 2.

03:07 For the first term here, we want to fit it into the form of f prime over fx.

03:18 And when you can fit that in, you can use the result on non -mode of fx...

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