Evaluate the integral. ∫(x^4+6 x^3+10 x^2+x)/(x^2+6 x+10) d x

Name: Problem 34, Chapter 7: Calculus Early Transcendentals
Uploaded: 2020-08-21T23:49:27-08:00
Duration: 5 min 1 s
Channel: Patrick Vaughn
Description: Evaluate the integral. ∫(x^4+6 x^3+10 x^2+x)/(x^2+6 x+10) d x

Evaluate the integral. ∫(x^4+6 x^3+10 x^2+x)/(x^2+6 x+10) d...

Key Concepts

Completing the Square

When dealing with quadratic expressions in the denominator that do not factor nicely, completing the square is a technique used to rewrite the quadratic in a perfect square form plus a constant. This transformation is particularly useful in integration as it prepares the expression for standard integral forms, especially those involving inverse trigonometric functions or substitutions that can simplify the integrand further.

Substitution Method

The substitution method, often referred to as u-substitution, is an integration technique used to simplify integrals by substituting a part of the integrand with a new variable. This technique is especially effective when the integrand contains a composite function whose derivative is present elsewhere in the expression. It transforms the integral into a simpler form that is easier to evaluate using standard techniques.

Integration of Rational Functions

This concept involves integrating functions that are expressed as the ratio of two polynomials. Often, the method starts by simplifying the rational expression using algebraic techniques to express it in a form that is easier to integrate, such as by performing polynomial division or partial fraction decomposition. Mastery of this concept is essential for handling integrals where the degree of the numerator is equal to or greater than the degree of the denominator.

Polynomial Long Division

Polynomial long division is a method used to simplify a rational function where the degree of the numerator is greater than or equal to the degree of the denominator. By dividing the numerator by the denominator, the rational function can be rewritten as a polynomial plus a remainder term over the original denominator. This step simplifies the integration process by separating the integrand into parts that are easier to integrate.

Transcript

00:01 In this problem, we want to integrate the function x to the fourth plus 6x cubed plus 10x squared plus x divided by x squared plus 6x plus 10 dx.

00:20 So by inspection, so we have, if you look at this term and you look at this term, notice that the top, is x squared times the bottom so let's go ahead and just break this fraction up to make it a little easier so we're going to have x squared times x squared plus 6x plus 10 over x squared plus 6x plus 10 plus x over x squared plus 6x plus 10 d x all right so this is going to cancel to just x squared and we're left with this.

01:15 So what do we do with this now? so if you'll recall how did you use completing the square.

01:22 So we want to put this in the form of x over something squared plus some constant squared.

01:32 So this is going to be like linear in x.

01:35 So like x plus something.

01:37 And this is going to be kind of like the leftover bits.

01:40 So let's go ahead and do that.

01:43 So if we had x, so what's one half of six? that's three.

01:48 So x plus three squared.

01:51 So that's x squared plus six x plus nine.

01:55 So if i take nine from the ten, i'll have one left over.

01:59 So i can write this as x over x plus three squared plus one.

02:12 Okay, so then if i, so what do we do with this now? so if u was equal to x plus 3 squared plus 1, then du by the chain roll, we're going to get 2x plus 3 dx.

02:37 So let's go ahead and do plus 3 minus 3.

02:43 So what does that do? so then i can break this fraction up again.

02:47 So we can have this piece with the x plus three over the x plus three squared plus one.

02:54 I can use a use substitution on that...

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