In Exercises 27–40, use a substitution to change the integral ...

Question

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ cos^(-1)(√x) / √x dx

Accepted Answer

Welcome back everyone. Evaluate the integral inverse of cosine square root of x. Dx divided by 2 square root of x. We're going to approach this problem using U substitution. Let's suppose that U is equal to the square root of x. Let's remember that the next step is to find the derivative of u with respect to x. That's the derivative of the square root of x, which is 1 divided by 2 square root of x. If we multiply both sides by dx, we can show that the d u is equal to dx divided by 2 square root of x, and this is exactly what we have within the integral, right? So now the integral becomes inverse of cosine of square root of x, which is now u, and dx divided by 2 square root of x becomes du u. Now let's remember the formula we can integrate inverse of cosine of x, d x as follows. Basically we are going to take x multiplied by inverse of cosine of. Acts in a general case. X is going to be our independent variable, right? We're not using any composite functions, so the answer will be x inverse of cosine of x. Minus Square root of 1 minus. X 2 plus a constant of integration c. So applying this formula we are going to get u because that's our independent variable. Inverse of cosine of u minus square root of 1 minus u2 + C. And now what is U? Well, it is square root of x. So we're going to replace you with square root of x. And we're going to get the square root of x inverse of cosine of. Square root of x. minus square root of 1 minus. U2, which is now square root of X2, and that's basically X. And that's how we arrived at the final answer. Let's label it and thank you for watching.

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.∫ cos^(-1)(√x) / √x dx

Watch next

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ cos^(-1)(√x) / √x dx