76. Different Substitutions
b. Show that ∫(1/√(x - x²)) ...

Question

76. Different Substitutions

b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x

Accepted Answer

Hello. In this video, we are going to be evaluating the given integral by using the given substitution, and we are going to assume that X is greater than or equal to 0. Now, in this problem, the integral that is given to us is the integral of X2 divided by the square root of X2 minus X to the 6th power DX. Before we approach this integral, let's just go ahead and simplify the integran first. Notice that within the square root we can factor out an X2d. If we were to factor out an X2d, we will be left with the integral of X2d divided by the square root. Of X2 multiplied by the quantity, 1 minus X 4th power. If we were to remove the X2d from the square root, that would simplify X2 to just be X. So we have the integral of X2 divided by X multiplied by the square root of 1 minus X the 4th power. And since we have a multiple of X in both the numerator and denominator, we can eliminate one of those multiples and allow us to simplify the integral to be the integral of X divided by the square root of 1 minus X to the 4th power DX. This is going to be the simplest form of the given integral. So now let's go ahead and approach this integral by using the given substitution. Now, the substitution that we are asked to work with is X is equal to the square root of U. Now, since we are substituting a value of X into terms of U, we need to come up with a substitution of D X in terms of you as well. The only issue here is that if we try to take the derivative of the current substitution, we're going to end up with some terms that we don't have in the integral. So instead, let's go ahead and rewrite the given substitution. If we were to square both sides of this substitution, we'll end up with U is equal to X squared. If we take the derivative of this substitution with retrospect to X, we will end up with DU is equal to 2 XD X. We have an XDX term within our integral, but we don't have this 2. So instead, if we were to divide two to both sides of this DU substitution, we end up with one half DU is equal to XDX. So when we apply this substitution, we are going to replace our XDX term with 1/2 DU. But what about this X to the fourth term within the integral? Well, we need to find a substitution for X to the 4th in terms of you. Now, going to U is equal to X2. If we were to square both sides of this U substitution, we will end up with X the 4th power is equal to U squared, and that is going to give us a substitution for X to the 4th in terms of U. So, by using XDX is equal to 12 DU, and the substitution for X to the fourth is equal to U squad, we can rewrite this integral to be the integral of 12 DU divided by the square root of 1 minus U 2 DU. Now, we will go ahead and pull the one half in front of the integral since it's just a constant function, and that is going to give us 1/2 multiplied by the integral of 1 divided by the square root of 1 minus U 2 DU. So, how do we simplify this integral now? Notice that 1 divided by the square root of 1 minus u squad matches the derivative of sine inverse. Recall that the derivative of sine inverse of X is equal to 1 divided by the square root of 1 minus u squared. Since the integrand matches the derivative of sine inverse, if we were to take the integral of that, we will end up with sine inverse of u. And so that means that the antiderivative of one divided by the square root of 1 minus u squared is going to be sine inverse of you. Now keep in mind that we still have this one half coefficient, so we are going to multiply sine inverse of you with this one half, and because we're working with an undefined integral, we have to add a generic constant C. So this is going to be the antiderivative of the integral, but keep in mind that we have to resubstitute you back into terms of X. Now, originally, we rewrote that substitution to be U is equal to X2. So by back substituting, the final version of this antiderivative is going to be 1/2 multiplied by sine inverse of X2 plus C. And this is going to be the solution to this problem, and the overall solution is going to be C. So I hope this video helps you in understanding how to approach this integral, and we will go ahead and see you all in the next video.

76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x

Key Concepts

Substitution in Integration

Inverse Trigonometric Functions

Definite vs. Indefinite Integrals

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