Evaluate the following integrals. ∫(x^4+1)/(x^3+9 x) d x |...

Name: Problem 58, Chapter 8: Calculus: Early Transcendentals
Uploaded: 2020-05-28T15:27:35-08:00
Duration: 7 min 14 s
Channel: Tom Greenwood
Description: Evaluate the following integrals. ∫(x^4+1)/(x^3+9 x) d x

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental algebraic tool that involves rewriting a polynomial as a product of its factors. In the context of rational functions, factoring the denominator can reveal opportunities for partial fraction decomposition and simplification of the integral.

Partial Fraction Decomposition

Partial fraction decomposition is a method of expressing a proper rational function as a sum of simpler fractions. Each of these fractions typically involves simpler denominators, such as linear or irreducible quadratic factors, which can be integrated using standard techniques.

Polynomial Division

Polynomial division is used when the degree of the numerator is greater than or equal to the degree of the denominator. This technique simplifies the rational function into a polynomial plus a proper rational function, which is easier to handle in integration.

Rational Function Integration

This concept involves integrating functions that are the ratio of two polynomials. Such integrals are often managed by first simplifying the expression either through polynomial division or decomposing the function into simpler fractions, making the integration process more straightforward.

Transcript

00:03 Okay, first thing you have to notice in this integral is that the exponent in the numerator is higher than the exponent in the denominator.

00:12 So we have to start this with some long division.

00:18 Bad part about this is you've got to fill in all those zeros to hold the places of all those terms inside your division sign.

00:32 And most of those we're not even going to need.

00:35 So we take x to the 4th divided by x to the 3rd and we get x.

00:41 By x, we get x to the fourth plus nine x squared.

00:48 Make sure you line up your like terms.

00:51 That's why we fill in all the zeros there, so that we don't leave out any terms.

00:57 And when we subtract those, the x to the fourths, subtract out, there's no x to the thirds.

01:02 We get negative 9x squared, and then we can bring down the plus one because that's our remainder, and we don't need to do any more dividing.

01:11 So this becomes the integral of x plus negative 9x squared plus 1 over the x cubed plus 9x.

01:26 If you don't mind, i'm going to factor out the x and make it x times x squared plus 9 dx.

01:36 And i did that because we have to have that factored version to do our partial fraction decomposition of this.

01:45 So fortunately, the decomposition isn't that bad.

01:51 We just have the two different fractions.

01:56 We have the linear factor of x.

01:59 So we're going to have an a over x.

02:03 And we have the quadratic factor, so that's got to have an x term and a constant term in the numerator.

02:09 So bx plus c over the x squared plus 9.

02:16 And so now we multiply all that by the common denominator.

02:19 So negative 9 x squared plus 1 equals.

02:26 When we multiply over here, each of the factors is going to divide out, and we have our numerator times the other factor.

02:34 So that would be a times x squared plus a times nine.

02:41 And then we'd have the bx plus c times the x.

02:44 So that's bx squared plus cx.

02:48 And now we can equate our terms and our coefficients.

02:54 So we've got our squared.

02:57 So negative 9 equals a plus b.

03:03 There are no x terms on the left hand side.

03:08 And just our cx on the right, so c is zero...

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