Evaluate the integrals in Exercises 69–134. The integrals are ...

Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

131. ∫ dx / (x√(1 − x⁴))

Accepted Answer

Welcome back everyone. In this problem we want to evaluate the indefinite integral of dx divided by x multiplied by the square root of 4 minus x 4. Now we can evaluate this integral using substitution. What do we know? Well, by substitution, OK, to make it easy for ourselves, let's first set U to be equal to X2. That means the derivative of u with respect to x is 2 x, so duu is going to be equal to 2 x dx, OK, which means then that dx divided by x will be equal to duu divided by 2 u, and that means we've now found a way to get a substitute for d x divided by x in our original problem in terms of u. So if we substitute it into the integral. This would imply then that the integral of d x divided by x multiplied by the square root of 4 minus x 4. Is going to be equal to half of the integral. Of duU divided by u multiplied by the square root of 4 minus u 2. Now again, we're in a bit of a problem. We'll need to use some more substitution to help us. So let's break this down with trigonometric substitution. So now let's let you be equal to 2 sine theta, OK, which means then that duu is going to be equal to 2 cu theta d theta, OK, so in that case we've had DU in terms of theta, we have you in terms of theta. Can we also express the square root of 4 minus u squad in terms of theta? Well, The square root of 4 minus u2 is going to be the square root of 4 minus 2 sin theta all squared in terms of theta. This will simplify to 4 minus 4 sin square theta. And now if we factor out 4 inside our square root, that's the square root of 4 multiplied by 1 minus sin squared theta. But if we recall, 1 minus sin squared theta equals cu squared theta by its identity. So really this is going to be the square root of 4 cus square theta, which will be equal to 2 costs theta. So no or integral of 12 multiplied by the integral of duu divided by u multiplied by the square root of 4 minus u 2 is going to be equal to a half of the integral of 2 cos theta d theta. Divided by 2 sine theta u multiplied by 2 cus theta, which was the square root of 4 minus u squared as we saw earlier, can we simplify this any further? Well, notice here if we cancel out our cost theta terms and 2, this simplifies to a half of the integral. Of d theta divided by 2 sine theta, and if we do some more simplifying, we can take 2 from our denominator and put it into our constant. In other words, that's a 1/2 multiplied by 12 or 1/4 and we'll be left with 1 divided by sine theta with respect to theta. But remember, 1 divided by sine theta is cosse theta. So this is really a quarter of cosse theta. OK, with respect to theater. Now, do we know anything about the integral of Corse theater with respect to theater? Well, we know. OK, if we have our quarter here. We know that the integral of Cosse theta is really the natural log of the absolute value of Cosse theta minus cut theta plus C, so this is our integral. Now that we have our integral, all we need to do now is to figure out a way to rewrite Cosic theta and cut theta in terms of X. How can we do that? Well, remember earlier we said that u equals x2 and U equals 2 sine theta. So here if I put this in blue. Since you. Equals 2 sine theta. OK, so we're starting with cosic theta to to rewrite it in terms of x. And since u equals 2 sine theta, then that means sine theta equals a half of u. And remember u is x 2, so that's really a half of x 2. Thus, if sine theta equals 12 of x2, then that means cos theta, it's reciprocal, will be equal to 2 divided by x2. So now this is a way, this is the first term that we've expressed in terms of x. Now we need to express cut theta in terms of x. What do we know about cut theta? Well, remember that the cotangent cut is equal to the or is the ratio of the cosine to the sine. So cut theta equals cost theta divided by sine theta. Remember, cost theta is the square root of 1 minus sine squared theta, and that helps because we already know that sine theta is x 2 divided by 2. So really then this is going to be the square root. Of 1 minus 12 of X22, OK. Divided by a half of x 2, which we know is sine theta, I know when we simplify we'll get 1 minus x4 divided by 4 under our square root, and in the interest of time when we simplify that it will become the square root of 4 minus x 4 divided by x square. So essentially we've now been able to rewrite cost theta. And sorry, Kosic theta and cut theta in terms of X. So if we put that back into our final integral. Then that means. A quarter of the natural log of the absolute value of Cosse theta minus cut theta plus c. is going to be equal to a quarter of the natural log of the absolute value of 2 divided by x2 minus the square root of 4 minus x 4 divided by x2 + c. And if we try to simplify what's inside our our absolute value here, OK. Then we'll end up with a quarter of the natural log of 2 minus the square root of 4 minus x4 divided by x2 since both of them have a common denominator of x2 plus c. Thus this is the integral of dx divided by x multiplied by the square root of 4 minus x 4. Thanks a lot for watching everyone. I hope this video helped.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.131. ∫ dx / (x√(1 − x⁴))

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Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
131. ∫ dx / (x√(1 − x⁴))