Reduction Formulae

Transcript

0:00Introduction

Hello everyone! Here is a small challenge first. Find the integral of tan x, multiplied by itself four times. No standard formula fits, right? And what if the power were ten, or twenty? Do we need a new trick for every power? Not at all. Today we build a ladder: one formula that brings the power down, step by step. These are reduction formulae, and they do appear in JEE problems. So, let's begin.

0:28The idea: a ladder for powers

So, first, a name. Capital I with a small n written low is read capital I sub n: the integral of tan to the power n x, d x. The long S is the integral sign, and d x means with respect to x. Tan to the power n x, with n written up high, means tan x, raised to the power n. Okay? Now, a big power is hard to integrate. So what is the next best thing? Yes, exactly! Link it to a smaller power. That is a reduction formula: capital I sub n written using capital I sub n minus one, or capital I sub n minus two. Right? Then we step down the ladder to capital I sub one or capital I sub zero, which are easy. These are indefinite integrals, so add plus capital C, the constant of integration, once, at the end. Clear?

1:25Building the formula for powers of tan x

Let's build the tan formula together. Split tan to the power n x as tan to the power n minus two x, times tan squared x; the dot means times. Now, which identity turns tan squared x into something friendly? Correct! Tan squared x equals sec squared x, minus one. Sec is secant, one over cos x, okay? Put that in and open the bracket. We get the integral of tan to the power n minus two x, times sec squared x, d x, minus a second integral, which is just capital I sub n minus two! With me? Now the first piece. Sec squared x is the derivative of tan x, so put t equal to tan x. Then we integrate t to the power n minus two, d t, and get tan to the power n minus one x, over n minus one. Since we divide by n minus one, n must be greater than or equal to two. Okay? So capital I sub n equals that fraction, minus capital I sub n minus two. Or the neat form: capital I sub n, plus capital I sub n minus two, equals tan to the power n minus one x, over n minus one. Got it? Now a quick one from the course. What is capital I sub five, plus capital I sub three? Just put n equal to five! Tan to the power four x, over four, plus capital C, the constant. No working at all. Nice, right?

3:05Powers of ln x: by parts with v=1

Next ladder: powers of log. Capital I sub n is now the integral of l n x, the natural log of x, in brackets, to the power n. Only one function, no product. So which method? Yes, exactly! By parts, with the v equals one trick from integration by parts. Log x, whole to the power n, is u, and one is v. Right? The integral of v is x. By the chain rule, the derivative of u is n times log x, whole to the power n minus one, times one over x. So we subtract the integral of x times that derivative. Okay? The x on top cancels the x below, and what is left gives n times capital I sub n minus one. So capital I sub n equals x times log x, whole to the power n, minus n times capital I sub n minus one. The power drops by one each step. Clear?

4:11Two more ladders from the course

Two more formulae from the course. For sine to the power n x: capital I sub n equals minus sine to the power n minus one x, times cos x, over n, plus n minus one over n, times capital I sub n minus two. It drops by two. So where does the ladder end? At capital I sub one when n is odd, and at capital I sub zero, the integral of one, which is x, when n is even. And a similar formula works for cos to the power n x. Got it? Last, for one over, x squared plus a squared, to the power n, where a is a constant: capital I sub n plus one equals x, over two n a squared times x squared plus a squared to the power n, plus two n minus one, over two n a squared, times capital I sub n. Okay? Now, careful here! This one climbs up, with n plus one on the left. So for capital I sub two, put n equal to one, not two. Clear?

5:19Example 1: integral of tan x to the power 4, two steps down

Now, our challenge from the start, as Example one. Using the reduction formula, find the integral of tan to the power four x, d x. Hint: two steps down the ladder. Pause the video and try it first. Done? Okay, let's check. Put n equal to four: capital I sub four plus capital I sub two equals tan cubed x, over three. So capital I sub four is that, minus capital I sub two. Okay? Next, n equal to two: capital I sub two plus capital I sub zero equals tan x, and capital I sub zero is x. So capital I sub two equals tan x minus x. Right? Now put it back: tan cubed x over three, minus the bracket, tan x minus x. So the answer ends in minus x, doesn't it? No, no, careful here! The minus sign before the bracket flips both signs: minus tan x, and plus x. Answer: one third tan cubed x, minus tan x, plus x, plus capital C, the constant. Perfect!

6:34Example 2: integral of (ln x) cubed, three steps down

Now a harder one, Example two. Find the integral of log x, whole cubed, d x. Hint: three steps down. Pause the video and try it first. Done? Okay, let's check. The log formula with n equal to three: capital I sub three equals x times log x whole cubed, minus three times capital I sub two. With me? With n equal to two: capital I sub two equals x times log x whole squared, minus two times capital I sub one. And capital I sub one, the integral of log x, you have seen before. What is it? Yes, exactly! x log x, minus x. Now we climb back up. Okay? Capital I sub two equals x times log x whole squared, minus two times that bracket: minus two x log x, plus two x. Right? Last step: capital I sub three is x times log x whole cubed, minus three times this capital I sub two. Watch the last term: minus three, times plus two x. What do you get? Minus six x, correct! And minus three times minus two x log x is plus six x log x. So capital I sub three equals x times log x whole cubed, minus three x times log x whole squared, plus six x log x, minus six x. Got it? Take x common in square brackets, and add plus capital C, the constant. See, the signs alternate, plus, minus, plus, minus, because of the minus n in the formula. Beautiful, right?

8:22Quick revision: tan and ln x ladders, the minus sign trap

So, quick revision. Write capital I sub n using a lower one, step down, then add plus capital C, the constant. Tan: capital I sub n plus capital I sub n minus two equals tan to the power n minus one x, over n minus one. Log: capital I sub n equals x times log x, whole to the power n, minus n times capital I sub n minus one. And a minus before a bracket flips every sign. Now go and try the practice problems on this page. All the best!