9–61. Trigonometric integrals Evaluate the following integrals...

Question

9–61. Trigonometric integrals Evaluate the following integrals.

59. ∫ from 0 to π/2 of √(1 - cos2x) dx

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says to evaluate the interval from 0 to pi divided by 4 of the square root of the quantity of 1 minus cosine of 4 X in quantity with respect to X. And so we need to evaluate this interval, and we're given the integral from 0 to pi divided by 4 of the square root of the quantity of 1 minus cosine of 4 X in quantity, and we're integrating this with respect to x. Now, in order to get this integral into a form that we can actually integrate, we need to use our trig identities. So we call that 1 minus cosine of 2 theta. It is equal to 2 multiplied by sine squared of data. So we're going to use this trig identity to rewrite our integral. So that means our integral is going to be equal to the integral from 0 to pi divided by 4. Of the square root of the quantity of 2 multiplied by sin squared of 2 X in quantity DX. So here the argument inside the sine squared function is going to be 2 X since we need to take half a 4, which is going to be 2. Now we can actually simplify this expression. So rewriting this, this is going to be equal to the interval from 0 to pi divided by 4 of the square root of 2, multiplied by the absolute value of the quantity of sine of 2 X. In quantity, DX. So here just, we've rewritten the square root of the sine squared function as being the absolute value of the sine function. Now if we look at the interval over which we're integrating over, we see that X is going to be between 0 and pi divided by 4. So that means that 2 X is going to be contained in the interval from 0 to pi divided by 2. And our sign function is actually positive over this entire interval. So this is in the first quadrant, and our sine function is positive in the first quadrant. So that means we can actually remove the absolute value signs out of our expression. So that means that our integral is going to be equal to the integral, going from 0 to pi divided by 4 of the square root of 2, multiplied by the sine of 2 X. DX. And so now we can actually perform the interval using substitution. So we're going to set U equal to 2 X, and that means that DU is going to be equal to 2DX. Now, we do not have 2 DX in our integral, so we'll just need to divide both sides of our expression for DU by 2, and this will give us 1 divided by 2, multiplied by DU is equal to DX. So now making these substitutions, we see that interval is going to be equal to the integral from 0 to pi divided by 2, since we need to change our variables here. And so, you is going to go from 0 to pi divided by 2. And then this is going to be um the interval of the square root of 2, multiplied by the sign of you. Multiplied by 1 divided by 2 multiplied by DU. And so now we can actually perform the integration. So the first thing we'll do is to pull out all of our constants in front of the integral sign. So this is going to be equal to the square root of 2 divided by 2, and what I'm left with is the integral of sine of U D U. and when we integrate the sine function, we get a minus cosine function. We'll have a minus cosine of you, and this is going to be evaluated at our end points of 0 and pi divided by 2. So, evaluating the endpoints, this is going to be equal to. The square root of 2, divided by 2, multiplied by the quantity of minus cosine of pi divided by 2. Plus the cosine of 0. And evaluating that, this is going to be equal to the square root of 2, divided by 2, multiplied by the quantity of minus, and the cosine of pi divided by 2 is 0, then we'll have plus the cosine of 0, which is just 1. And finally evaluating this expression, this is just going to be equal to the square root of 2, divided by 2, and this here is the answer that we're looking for for this problem. So I hope this explanation has been useful, and I'll see you in the next video.

9–61. Trigonometric integrals Evaluate the following integrals.
59. ∫ from 0 to π/2 of √(1 - cos2x) dx

Key Concepts

Trigonometric Identities

Integration Techniques

Definite Integrals

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