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.

∫ x / (1 + √x) dx

Accepted Answer

Find the antiderivative of the square root of x divided by 1 plus the square root of xDX. Now, to solve this, we're going to use a substitution that makes this easier to handle. So, Let's first use our substitution U equals 1 + 1 square root of x. Now the substitution can be a little tricky, but we're going to solve this 4 square of X, to get the square root of X equals u minus 1. We also need a DU. So DU is going to be the derivative of U. Which is just 1 divided by 2 squares of x. We can solve for DX. To get DX equals 2 squares of X. Do you We can also use a substitution for a square root of X, which is u minus 1. Which means we get 2 multiplied by u minus 1 DU. Now, we can plug all of our values into our original integral. We get the integral Of u minus 1. Divided by you, multiplied by 2 multiplied by u minus 1. Which simplifies to 2 integral of u minus 1 squared divided by u. Now, the easiest way to solve this is to factor U minus 12, to give us U2 minus 2U plus 1, all divided by U. Finally, we'll divide every term and enumerator by u to get 2 integral of u. -2 + 1 divided by UDU. Now from here, we can find the integral of every one of our terms based on integral rules. We have 2 multiplied by the integral of u or antiderivative, which is u square. Divided by 2. Minus 2 U. Plus the integral of one divided by you which is natural log absolute value of you. We will just add a constant K at the end to account for the indefinite integration. Now from here, we can resubstitute you and distribute R2. We get u2 minus 4U plus 2 natural log absolute value view, which simplifies when we plug in our u. To give us 1 + square root of x 2. -4 multiplied by 1 + square of x + 2 natural log absolute value of 1 + square of x. Plus C at the end. Now from here, we can simplify even further. For to simplify our constants, any one of them that we have. To give us Our new answer. This is 1 + 2 square of X. Plus X minus 4 minus 4 squares of x + 2. Natural log of 1 plus the square of x + k. Whenever we expand all of our terms, and we'll leave this as K for now, actually. From here we can simplify once again. To give us X minus 22 x + 2 natural log, 1 + 2 of x. Any of our coefficients remaining ink, we will all simplify to just give us one constant of C. This is the final answer to. Our antiderivative. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

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

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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.
∫ x / (1 + √x) dx