5-32 Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. ∫0^π x^3 sinx

step 1 The key concept for this problem is integration using tables. And the first steps whenever we're

5-32 Use the Table of Integrals on Reference Pages 6-10 to...

Key Concepts

Definite Integration

Definite integration involves calculating the integral over a specific interval, which yields the net area under a curve between the given bounds. In this context, it is necessary to evaluate the antiderivative at the upper and lower limits and subtract to obtain the final value.

Integration by Parts

Integration by parts is a key technique used to integrate products of functions, based on the product rule of differentiation. It breaks down a complex integral into simpler parts by designating one function to be differentiated and another to be integrated, making it particularly useful when dealing with products of polynomials and trigonometric functions.

Reduction Formula

A reduction formula is a method that expresses an integral with a given exponent in terms of another integral with a lower exponent. This recursive approach simplifies the process of integrating functions involving powers, especially when combined with trigonometric functions as encountered in many calculus problems.

Table of Integrals

A table of integrals is a compiled resource of standard integral forms that can be referenced to quickly evaluate integrals without performing each integration step manually. Utilizing such a table can save time and reduce errors when integrating common forms, as it provides formulas that directly apply to the integral in question.

Transcript

00:01 The key concept for this problem is integration using tables.

00:06 And the first steps whenever we're doing integration using tables is to either make a substitution so we can find the right table entry or sometimes we can find the right table entry right away.

00:17 So the problem we're going to work on today is the integral down here in black, the integral from zero to pi of x cubed sine of xdx.

00:27 And if you're looking at your tables, you'll notice there is a nice integral up here in red, the integral of u to the n sign u, which matches really nicely.

00:40 Our x is u and our n is three.

00:43 So it matches the one in red perfectly.

00:47 The reason the one in blue is up here also is if you look at the integral in red, the table entry, it's one of those iterative entries.

00:55 So you have an integral as part of the answer.

00:59 So we're going to have to pop back and forth between the red and the blue.

01:03 And we'll show you how to do that here in just a second.

01:07 But again, down here with the black integral that we're working on, we have limits from zero to pi.

01:14 And again, we're going to just worry about those at the very end.

01:17 We'll plug those in.

01:18 But again, my u is x.

01:21 So that's great.

01:23 D .u is just dx.

01:26 So there's nothing to really worry about with the substitutions.

01:30 I can just use use instead of the x's.

01:33 And then my n to start off with, the power on the x is 3.

01:40 Okay, so i'm going to start off using the red integral here.

01:45 So this is equal to, using the red integral, negative, x is you again.

01:54 So we can just write x cubed cosine x plus n is three.

02:02 So three times the integral of x to the three minus one.

02:09 So x squared cosine x d x.

02:17 But now that answer has an integral in it.

02:22 So i have to be able to do the integral of x squared cosine x.

02:25 And that one is going to use the blue formula here.

02:31 So this integral is now equal to, let's make sure and copy down the first part here, the negative x cubed cosine x plus three.

02:46 And now to do this integral, i'm going to use the blue formula, because again, it is a u to the n cosine u -d u.

02:57 And again in this case n is 2 okay so we're going to do x squared sine of u but x is just you so x squared sine x minus now n is 2 so 2 times the integral of u is x to the n minus 1 so 2 minus 1 is 1 sine of u d u and again u is x so let's just make that switch again so again x and u the same thing in this problem great but we're still not done we still have an integral here and so that integral is going to be the red equation again because it's an x to the first sign of x so it's going to look like the red equation again with n equaling 1.

04:08 And so let's again, let's copy down everything we have so far.

04:12 Negative x cubed, cosine x, plus 3 times the integral, x squared sine of x minus 2 times the integral.

04:30 And now we're going to use the red formula one more time with n being 1.

04:36 So we're going to have negative x to the first power, cosine u, but again, u is just x, plus n is one still this time.

04:51 So up here, n is one.

04:56 So we're going to do the one times the integral of x to the 0th power, cosine x dx.

05:13 Super.

05:14 Now we're almost done with the integrating part.

05:17 We still have to worry about our limits, but this x to the zero here, we can simplify that because x to the zero is one.

05:28 So i'm just going to erase that, okay? because again, x to the zero is one.

05:37 And so now i just have one times cosine of x, which is just a cosine of x.

05:42 Okay, and so that's a basic integral that i can just do.

05:48 All righty.

05:50 So to finish this out here, the integral is equal to negative x cubed cosine x plus three times, make sure we're getting our parentheses straight here, x squared sine of x minus two times negative x cosine x plus, and now the integral of the sign of x, that's one of our basic.

06:23 Integrals.

06:24 That's just the integral of cosine of x is the sign of x...

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