Evaluate the integrals in Exercises 1-22 . ∫0^π/ 2 sin^2 2 θcos^3 2 θd θ

Name: Problem 22, Chapter 8: Calculus
Uploaded: 2020-09-01T20:05:50-08:00
Duration: 10 min 9 s
Channel: Prakash Hampole
Description: Evaluate the integrals in Exercises 1-22 . ∫0^π/ 2 sin^2 2 θcos^3 2 θd θ

Evaluate the integrals in Exercises 1-22 . ∫0^π/ 2 sin^2 2...

Key Concepts

Trigonometric Identities

Trigonometric identities, such as the double angle formulas and power-reducing formulas, are essential tools for rewriting integrals that involve products or powers of trigonometric functions. These identities allow us to express the integrand in a simpler form that is easier to integrate.

Substitution Method

The substitution method is a key integration technique used to simplify the integrals by changing the variable. In the context of trigonometric integrals, it often involves setting a new variable equal to a trigonometric function (like sin or cos) whose derivative is present in the integrand, thereby transforming the integral into a standard form.

Integration of Powers of Trigonometric Functions

When faced with integrals of trigonometric functions raised to a power, strategies vary depending on whether the power is even or odd. In cases with odd powers, one typically isolates one trigonometric factor and uses a substitution based on its derivative, while even powers might require applying power-reducing identities.

Changing Limits of Integration

For definite integrals, after applying a substitution, the limits of integration must be adjusted to correspond to the new variable. This step is crucial for ensuring that the evaluated integral accurately reflects the area or value being measured over the new interval.

Transcript

00:01 When we have powers of sign and coasts like this, we usually use the, use identities derived from the double angle formula to rewrite them as follows.

00:10 So i'm going to take a cos 2 theta out, just going to put that at the beginning.

00:16 I'm going to rewrite the sine squared 2 theta as half of 1 minus cos 4 theta.

00:26 And i'm going to rewrite the remaining co -square 2 -theta as half of 1 minus cos 4 -theta, sorry, plus 4 -theta plus course 4 -theta.

00:47 From here, i'm going to take the half and the half outside.

00:52 I'm going to put them at the beginning.

00:55 So we have 1 -4th cost 2 -theta times this.

01:00 You may notice this is a difference of squares.

01:08 So when we multiply the two things that are in brackets here, we get 1 minus co -squared for theta.

01:24 Now for that co -squared there, i'll actually first...

01:30 Yeah, okay, for that cos squared there, i'm just going to...

01:34 I will rewrite that again using that formula.

01:43 In fact, i'm just going to make it clear.

01:46 What i'm using here is cost squared x equals half of 1 plus cos 2x.

01:56 So here we have half, half of 1 plus.

02:05 Now i'm going to double the 4theta to get kos 8 theta.

02:17 I'm going to expand that negative half.

02:21 And we have 1 minus half, minus.

02:25 Half cos 8 theta 1 minus half is 1 half we can factor out the half from the brackets you have 1 eighth of um 1 minus uh coast 8 theta and um that can be split into sorry i just realized dumb don't make this don't make a mistake like this um there is still this coast 2 theta here so i'm going to just make sure i put that back in.

03:34 Now what i'm going to do is i'm going to take this constant multiple 1 eighth outside, and then we have the integral of this.

03:42 I'm going to do distributive property.

03:45 We have cos 2 theta when you multiply by the 1, and then minus cost 8th theta times cos 2 theta.

04:04 And by subtraction rule, this would be equal, this can be split up into these two integrals.

04:12 But make sure to put them in brackets so the whole thing gets multiplied by the 1a.

04:18 And from here, for that first integral, well, that first integral will be easy enough to find.

04:28 But for the other one, we're going to have to use an identity, which is kosemx.

04:39 Times cos n x is equal to half of uh...

04:43 Coast m minus x sorry m minus n x plus cos m plus n so okay first for the that first integral you have um coast we have the integral of uh coast 2 theta which is uh we'll make sure to divide by the 2 and the integral of cos is sign so half sine 2 theta then for the other integral, we have half of this...

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