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.

125. ∫ dx / (√x * √(1 + x))

Accepted Answer

Welcome back everyone to another video. Find the integral dx divided by the square root of x, square root of 9 plus x. For this problem we're going to use substitution. Let's suppose that square root of x is equal to u. Now since we have x as well within the integral, we're going to show that x is equal to u squared if we square both sides, right? So now from here, let's evaluate Dx divided by DU, which is the derivative of X, and that's to you using the power rule. Derivative of U squared is 2U, right? And now what we're going to do is solve for DX. DX is equal to 2U multiplied by DU. So now the integral can be written as dx, which is 2 u multiplied by d u divided by the square root of x which is u square root of. 9 + x, which is 9 + u squared. Notice that we can cancel out u. We can also factor out the constant to get 2 integral d u divided by. Square root of? 9 plus u squared and this is one of the integrals that we know. Specifically, let's remember that the integral of d u divided by a square root of A 2 + u2 is equal to inverse of sin. Of u divided by a plus c and in this case a is equal to 3 because 3 squared is 9 so we're going to get 2 multiplied by inverse of sin. Of u divided by a, which is 3 plus c. Finally, let's replace you with the square root of x to get two inverse of cinch. The square root of x divided by 3 plus a constant of integration c and that's our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.125. ∫ dx / (√x * √(1 + x))

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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.
125. ∫ dx / (√x * √(1 + x))