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

Question

9–61. Trigonometric integrals Evaluate the following integrals.

60. ∫ from 0 to π/8 of √(1 - cos8x) dx

Accepted Answer

Hi everyone, let's take a look at this practice problem. This prompts us to evaluate the integral from 0 to pi divided by 10 of the square root of the quantity of 1 minus cosine of 10 X in quantity with respect to X. So we need to evaluate the interval from 0 to pi divided by 10 of the square root of the quantity of 1 minus cosine of 10 X in quantity, and we're integrating that with respect to X. Now, in order to get this integral into a form that we can actually integrate, we need to use some trig identities. So recall that 1 minus cosine of 2 theta. is equal to 2 multiplied by the sin squared of data. So we can use this identity to rewrite our integral. So that means our integral is going to be equal to the integral from 0 to pi divided by 10. Multiplied by the square root. Of the quantity of 2 multiplied by sin squared of 5 X in quantity, and that's multiplied by DX. Now, here we have 5 X as the argument, since we need to take half of 10, which will give us 5. And we can rewrite this expression as being equal to the interval from 0 to pi divided by 10. Of the square root of 2. Multiplied by the absolute value of sine of 5 X. DX. So, here we've just rewritten the square root of the sine squared function as being the absolute value of that sine function. Now we're performing this interval, this interval over the interval of X going from 0 to pi divided by 10. So that means that 5 X is going to be contained in the interval, going from 0 to pi divided by 2. And our sign function on the interval from 0 to pi divided by 2 is going to be positive, so we can actually remove the absolute value um signs, since our sign function is going to be positive over this interval anyways. So that means that we can rewrite our integral as being equal to the integral from 0 to pi divided by 10. Of the square root of 2 multiplied by the sine of 5 X D X. So now we can perform this interval by using substitution. So we'll set U equal to 5 X. And so that means that DU is going to be equal to 5 DX. And if we look at our integral, we only have DX, so we're going to take our DU expression and divide both sides of that equation by 5. And so we'll get 1 divided by 5 multiplied by DU is equal to DX. So now we can make these substitutions into our integral. In doing so, this is going to be equal to the integral, and here we have to change our limits of integration, and the limits are going to go from 0 to pi divided by 2. And then we'll be integrating the square root of 2 multiplied by the sign. Of you Multiplied by 1 divided by 5, DU. So now we just need to perform the integration, so this is going to be equal to. We'll pull all of our consonants out in front of the integral, so we'll have the square root of 2 divided by 5. And this is going to be multiplied by the quantity, and here we're going to be integrating the sign of view with respect to you. Recall that when you integrate sine of view, you get minus cosine of you. And this is going to be evaluated at our end points of 0 and pi divided by 2. So, now, evaluating our end points, this is going to be equal to. The square root of 2, divided by 5, multiplied by the quantity of minus cosine of pi divided by 2. Plus the cosine of 0. And evaluate this expression, this is going to be equal to the square root of 2 divided by 5, 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 then finally, evaluating this expression, this is just going to be equal to the square root of 2, divided by 5. And this here is the answer that we are 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.
60. ∫ from 0 to π/8 of √(1 - cos8x) dx

Key Concepts

Trigonometric Identities

Integration Techniques

Definite Integrals

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