Two integration approaches Evaluate ∫cos(lnx) d x two...

Key Concepts

Integration by Substitution

Integration by substitution is a method where the integrand is transformed using a change of variable. This technique simplifies the integral by replacing a complex expression (often involving a composite function) with a single variable, provided the differential also changes accordingly. It is particularly useful when the integrand contains a function and its derivative, as it transforms the integral into a more standard or recognizable form.

Integration by Parts

Integration by parts is a fundamental technique based on the product rule for differentiation. It allows the integration of products of functions by choosing parts of the integrand to differentiate and integrate. This method involves applying the formula ? u dv = u*v - ? v du, and it may need to be used iteratively when the resulting integral still contains a product of functions. In some cases, repeated application leads to an equation that can be solved for the original integral.

Use of Integral Tables

Integral tables are compilations of standard integrals that provide direct formulas for commonly encountered functions. They are useful resources when an integral, after the appropriate substitution or manipulation, matches a standard form listed in the table. By referencing these tables, one can bypass lengthy computations and quickly find antiderivatives for a variety of functions.

Transcript

00:01 Section 7 .6 problem 76.

00:04 So here they give us an integral cosine of the natural log of x, and they want us to solve this two different ways.

00:11 First, they want us to substitute u as equal to the natural log of x, and then use an integral table, and then secondly, they want to use integration by parts.

00:22 Okay, so the first thing we want to do, let's do the substitution that they require.

00:26 So they're asking us to let u equal the natural log of x, so that means that du is equal to 1 over x, dx.

00:39 So that tells me that dx is equal to what x, du? and you know that if u is equal to the natural log of x, then e to the u is equal to x.

01:02 So this tells me that dx is equal to e to the u, du.

01:11 So my substitution here is u and this is d x.

01:18 So this integral transforms into the integral of the cosine.

01:24 So natural log of x, that's just u, and then times e to the u, to you.

01:36 So i've got the, i need to figure out the integral of e to the u, cosine of u, d u, and this one you should be able to find easily in an integral lookup table.

01:50 So this is just one half x and then the sign of the natural log of x plus a cosine of the natural log of x plus a constant of integration.

02:15 So that is our solution by using the use substitution.

02:20 Now that's part a of this problem.

02:22 Part b, they say if you did not know that substitution, could you use integration by parts to get to the same spot? so if i started out with the integral of what was it, the cosine of the natural log of x, dx, could i use integration by parts and get to the same answer? hopefully that answer is yes.

02:46 So let's let you equal the cosine of the natural log of x, then du.

02:57 So when you take the derivative of cosine, you end up with minus sign...

Please give Ace some feedback