Evaluate the integrals in Exercises 1-22 . ∫16 sin^2 x cos^2...

Key Concepts

Integration Techniques for Trigonometric Functions

Integrating trigonometric functions often involves rewriting the integrand using identities to simplify the expression. By combining the double-angle and power reduction formulas, one can break down complex trigonometric integrals into a sum of simpler integrals that are straightforward to evaluate.

Double-Angle Identity

The double-angle identity, such as sin(2x) = 2 sin x cos x, is used to simplify products of sine and cosine functions. By expressing the product sin²x cos²x in terms of sin²(2x), the integral is transformed into a form that is easier to integrate.

Power Reduction Formula

The power reduction formula converts trigonometric functions raised to a power into expressions involving first powers, typically by using identities such as sin²? = (1 - cos(2?))/2. This technique simplifies the integration of functions that involve squared trigonometric terms.

Transcript

00:01 Alright, so we have this integral of 16 sine squared x, cosine squared x, dx.

00:08 And we'll notice that we have both even powers of sine and cosine.

00:13 So i feel like either method works, but you can convert your signs to cosine, to your cosines to signs, and you should have the same process and essentially the same process, and then end up with the same method at the end.

00:28 So i'm going to go ahead and turn our signs into cosines.

00:33 Using our identity, sine squared x is simply 1 minus cosine squared x.

00:40 And we have the cosine squared x d x as follows.

00:44 And if we go ahead and pull out the 16, we end up with an integral of cosine squared x minus cosine to the fourth x d x.

00:53 And these are two integrals that we can do.

00:56 So we have our cosine to the fourth.

00:58 Which is double power reduction and a simple power reduction for this cosine.

01:04 So if we go ahead and first, we can first separate this into 16 times the integral of cosine x dx, and then minus 16 times the integral of cosine to the fourth dx.

01:19 So we're going to go ahead and solve this problem separately, just so we can go ahead and do one integral at a time instead of worrying it all about it at once.

01:28 So our first integral right here on the left is the easier one to do.

01:33 So it's just a simple power reduction.

01:35 We can go ahead and turn cosine squared x into 1 plus cosine 2x over 2.

01:41 And the integral of this, we'll bring the 1 half out and take the integral of 1 plus cosine 2x the x, simply giving us 1 half x plus sine 2x, 2x over 4.

02:01 And then your c's, you don't need your c just yet because we're doing a simple integral, which is part of a bigger answer.

02:08 So we have our first integral evaluates to this...

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