The integrals in Exercises 1–44 are in no particular order. Ev...

Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx

Accepted Answer

Welcome back everyone. Compute the integral 2 x plus 3 square root of x minus 2 dx divided by 2 x square root of x minus 2. For this problem, let's go ahead and rewrite the integral as follows. We're going to perform division. So first of all, there'll be integral of 2 x divided by the denominator 2x. square root of x minus 2 Followed by D X. And for the 2nd integral, we're going to add. Integral 3 square root of x minus 2 divided by 2 x square root of X minus 2 D X Notice that this is really helpful because we can cross out multiple terms. For the first integral, we're going to cross out 2 x and for the second integral we can cross out the radical part. And now we end up with two integrals. For the first one, we have d x divided by square root of. X minus 2. For the second one we can factor out the constants that be 3 divided by 2 or 3 halves integral. D X divided by x. Well then, and now let's focus on each integral separately. For the first integral, we're going to use U substitution. Let's suppose that U is equal to x minus 2. And then the derivative of u with respect to x is going to be the derivative of x minus 2, which is 1, right? So that from here we can show that duu is equal to dx and we can rewrite the integral as duu. Divided by the square root of u and now how do we integrate. Well, we can use the power rule. We can rewrite 1 divided by the square root of you as you raise the power of -1/2. And integrate from here so that be u to the power of -1. 1/2 + 1 divided by -1 1/22 + 1. Plus a constant of integration C1. Which is equal to u to the power of 1/2 divided by 1/2, which gives us 2 in front plus c1, and we know what u is. So we basically get 2 u to the power of 1/2, which is X minus 2 to the power of 1/2 and that square root of x minus 2. Plus C1. So we can substitute this value to square root of. X minus 2 for now we can ignore the constant of integration c 1 and for the second integral, let's remember that the integral of d x divided by x is an elementary integral. Its value is ln of the absolute value of x. And we're going to add the constant of integration C too. Now, there's no necessity to write the absolute value because according to the domain of the function, x must be greater than 2, so X is positive. Meaning the answer for this problem is going to be 2 square root of x minus 2+. Three halves. Ellen Of x plus c, where c is a combination of constants C1 and C2, that's our final answer, and thank you for watching.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx

Watch next

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx