Evaluate the following integrals. ∫0^2 (x+2)/(x^2+4 x+8) d x...

Key Concepts

Rational Function Integration

This concept involves techniques used to integrate functions that are ratios of polynomials. In many cases, especially when the degree of the numerator is lower than that of the denominator, the integral can be tackled using methods such as substitution or partial fractions after simplifying the expression appropriately.

Substitution Method

The substitution method is a standard technique in integration where a new variable is introduced to simplify the integrand. In integrals involving rational functions, a careful choice of substitution can transform the integrand into a form that is easier to integrate, often reducing it to a standard integral form.

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in a form that reveals certain inherent symmetries or standard forms. This method is particularly useful in integrals where the quadratic expression appears in the denominator, as the rewritten form often facilitates the application of substitutions or integration formulas such as those leading to logarithmic or inverse trigonometric functions.

Transcript

00:01 Section 7 .1, problem 34, we're dealing with an introduction to techniques of integration here.

00:07 So typically when you see a quadratic in three terms like this, the key is going to be completing the square.

00:14 So let's look at this x squared plus 4x plus 8.

00:20 In order to complete the square, what is half of four? that's two, you square that you get four.

00:27 So i could write it like this.

00:29 But more importantly, that gives me x plus 2.

00:32 Squared plus 4 so let's work on just the indefinite integral first so the indefinite integral of x over it's going to be written as x plus 2 squared plus 4 dx so substitution would be let u equal x plus 2 then d u is just going to be d x and we we also know that x is going to be u minus 2.

01:11 So this integral turns into replace x with u minus 2 and then you're going to have u squared plus 4 du.

01:26 So how do i work on this? let's break this into two separate integrals.

01:30 So i could say this is the integral of u over u squared plus 4 d u and then minus 2.

01:40 The integral of 1 over u squared plus 4 d u so on the left -hand side let's make a substitution that if z is equal to u squared plus 4 then d z is equal to 2 u d u d u and so to transform this if i had a 2 in front and a one -half then this would turn into the integral one -half, 2u -d -u, that's d -z, so this is just 1 over z, d -z, and you know that that is a natural log, so this just turns into one -half, natural log, absolute value of z, plus a constant of integration.

02:40 What is z? z is u -square plus 4, so this is 1 -half natural log, u -squared plus 4 and that's always positive so i don't need absolute value signs what is u squared plus 4 well u squared plus 4 that is this original expression that you see right here so x squared plus 4x plus 8 so this turns into one -half log x squared plus 4x plus 8 plus a constant integration so that's the anti -derivative of the first half.

03:21 And now let's look at the next one.

03:22 This one is just much simpler.

03:24 This is an arc tangent.

03:25 So minus 2.

03:26 And this is going to be 1 over 2, arc tangent of x over 2.

03:35 And so this is just minus arc tangent of, excuse me, not x but u.

03:43 Sorry about that.

03:51 So plus a constant...

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