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

Question

9–61. Trigonometric integrals Evaluate the following integrals.

57. ∫ from 0 to π of (1 - cos2x)³ᐟ² dx

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This promises to evaluate the interval from 0 toi of the quantity of 1 plus cosine of 2 X in quantity rates to the 3 half power with respect to X. So we're asked to evaluate this interval, and we're given the integral from 0 to i of the quantity of 1 plus cosine of 2 X. In quantity raised to the 3 house power, and we're integrating this with respect to X. Now, in order to get this into a form that we can actually integrate, we need to use some trick identities. So we call that 1 plus cosine of 2 X. It is equal to 2 multiplied by cosine, squared of X. So using this identity, this means that integral is going to be equal to the integral of 0 to pi. Of the quantity of 2 multiplied by cosine squared of X in quantity raised to the 3 has power DX. And we can actually simplify this expression. This is going to be equal to the integral from 0 to pi. Of 2 raised to the 3 has power, multiplied by the absolute value of the quantity of cosine cubed of X in quantity DX. Now, this function has even symmetry about X equal to pi divided by 2. So that means that we can rewrite this integral as being equal to. 2 multiplied by the integral from 0 to pi divided by 2, of 2 raised to the 3 has power, multiplied by the absolute value of the quantity of cosine cubed of X in quantity DX. Now, since we're integrating over the interval from X equal to 0 to pi divided by 2, our cosine function is actually positive over this interval, so we can remove the absolute value signs. And that means that this is going to be equal 2, and here we'll take the opportunity to combine all of our our consonants out in front of the integral. So we'll have 2 multiplied by 2 to 3 has power, which gives me 4 multiplied by the square root of 2, and this gets multiplied by the interval from 0 to pi divided by 2 of cosine cubed of XDX. Now what we want to do now is rewrite cosine cubed of X as being equal to cosine of X multiplied by cosine squared of X. So doing so, this is going to be equal to 4 multiplied by the square root of 2, multiplied by the integral, from 0 to pi divided by 2. Of cosine of x multiplied by cosine square of X, which we will rewrite as 1 minus sine squad of X in quantity DX. So here we use the trig identity that cosine square of X, plus sine square of X is equal to 1. And so now we just need to distribute the cosine term inside the integral. And so this is going to be equal to 4 multiplied by the square root of 2. Multiplied by the quantity of the integral from 0 to pi divided by 2 of cosine of X D X. Minus the integral from 0 to pi divided by 2 of squared of X multiplied by cosine of XDX. In quantity. So, the first integral here, we can actually just integrate straight as a straightforward integral. But for the second integral, we actually need to use use substitution. So we're gonna set you. Equal to sin of X. And that means that DU is going to be equal to cosine of X D X. So the integral from 0 to pi divided by 2 of sine squared of X multiplied by cosine of X. DX Becomes equal to the integral, and here we do have to worry about our limits of integration. So when X is equal to 0 for our lower limit, we'll have the sign of 0, which is going to be equal to 0. So our lower limit in terms of you is going to be equal to 0. And when X is equal to pi divided by 2 for upper limit, we will have the sign of pi divided by 2, which is equal to 1. So our upper limit for you is going to be equal to 1. So our intervals could go from 0 to 1. And inside the integral we'll have U squared multiplied by DU. Now, this interval is easy enough to integrate, but this is going to be equal to. When I integrate U2d, I get U cubed divided by 3, and we'll need to evaluate this at the limits of 0 to 1. And doing so, this is going to be equal to 1 divided by 3 minus 0 divided by 3, which is equal to 1 divided by 3. So, with that information, we see that our original interval, which is going to be the integral. From 0 toi of the quantity of 1 plus cosine of 2 X in quantity raised to the 3 has power DX. is going to be equal to. 4, multiplied by the square root of 2, multiplied by the quantity, and here we'll have to integrate that first integral. So when we integrate cosine of X with respect to X, we get sin of X. And we'll need to evaluate that at the end points of 0 and pi divided by 2. Then we'll have minus, and here we'll have the integral from 0 to pi divided by 2 of sin square of X multiplied by cosine of XDX, and we already evaluated that integral to be 1 divided by 3 in quantity. So evaluating this expression, this is going to be equal to 4 multiplied by the square of 2, multiplied by the quantity. And here when we evaluate our upper and lower bounds, we see that the sign of pi divided by 2 for the upper limit is going to give us 1. Then we'll have minus the sign of 0 for the lower limit, which is going to give us 0. And then we'll have minus 1 divided by 3 in quantity, and evaluating this expression, this is going to be equal to. 8, multiplied by the square of 2, divided by 3, and this here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

9–61. Trigonometric integrals Evaluate the following integrals.
57. ∫ from 0 to π of (1 - cos2x)³ᐟ² dx

Key Concepts

Trigonometric Functions

Integration Techniques

Definite Integrals

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