Integral

[lilgnome57]

1. What is the integral of (6x-x^2)^-1/2)dw using the substitution of u=x^1/2? PLease walk me through how to derive the integral

2. What is the integral of ((sqrt(x)+x)^-1) using substitution?? Please walk me through this one as well.

3.Also, how would you go about proving that the Simpson's rule is exact when approximating the integral of any cubic polynomial function?

Thank You!

 

[galactus]

\(\displaystyle \int\frac{1}{\sqrt{6x-x^{2}}}dx\)

Do you HAVE to use that substitution?. There are easier ones.

Try \(\displaystyle x=u+3, \;\ dx=du\)

This leads to \(\displaystyle \int\frac{1}{9-u^{2}}du\)

Now, make the sub \(\displaystyle u=3sin(w), \;\ du=3cos(w)dw\)

and it whittles down to practically nothing to integrate.

\(\displaystyle \int\frac{1}{\sqrt{x}+x}dx=\int\frac{1}{\sqrt{x}(1+\sqrt{x})}dx\)

Let \(\displaystyle u=1+\sqrt{x}, \;\ 2du=\frac{1}{\sqrt{x}}dx\)

 

[lilgnome57]

Thank you,
But actually yes, for the first integral i must use the substitution of u=sqrt(x). Could you help me do this?

 

[galactus]

Okey-doke.

If \(\displaystyle u=\sqrt{x}\), then \(\displaystyle x=u^{2}, \;\ dx=2udu\)

Sub \(\displaystyle u^{2}\) in place of x and \(\displaystyle 2udu\) in place of dx.

Then, simplify it down. You will get an integral similar to the one in my last post.

Let me know how you progress.