Integrals on exam I couldn't answer

1. Dec 1, 2003

noboost4you

On a recent exam I took, these integrals came up and I was unable to answer them correctly:

$$\int \frac{2}{(x-1)^2+1} dx$$

and

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

I had absolutely no idea how to solve those 2 equations and my professor decided not to go over any of the problems from the test. He did mention that I needed to use substitution, but I am still unclear of how to solve them.

Can anyone help me out?

Thanks

2. Dec 1, 2003

Ambitwistor

Re: integrals

Do you know how to do these integrals?

$$\int \frac{1}{x^2+1}\,dx$$

and

$$\int \frac{1}{\sqrt{1-x^2}}\,dx$$

3. Dec 1, 2003

noboost4you

Re: Re: integrals

$$\int \frac{1}{\sqrt{1-x^2}}\,dx$$ = inverse sine

and

$$\int \frac{1}{x^2+1}\,dx$$ = inverse tangent

I only know those answers by rule, when he added more constants and suggested substitution is when I didnt understand.

4. Dec 1, 2003

Ambitwistor

Re: Re: Re: integrals

The factors of 2, you can just pull out of the integral, since they're constant multiplicative factors. As for the substitution, if you'd like to turn your integrals into the ones I gave, then you can see that you'd really like there to be x's where you have (x-1)'s. So that's what you have to substitute x = x-1. To be clear, introduce a new variable: u = x-1. Then du/dx = 1, so du = dx, and you can rewrite the integral purely in terms of u by substituting u for x-1 and du for dx.

5. Dec 1, 2003

noboost4you

Re: integrals

$$\int \frac{2}{(x-1)^2+1} dx$$ would equal

2$$\int \frac{1}{(x-1)^2+1} dx$$
then u = x-1
du = dx
the new integral then equals
2$$\int \frac{1}{u^2 +1} du$$
which in turn equals 2tan^-1(x-1) ??

and

$$\int \frac{2}{\sqrt{1-(x-1)^2}} dx$$ would equal

2$$\int \frac{1}{\sqrt{1-(x-1)^2}} dx$$
then u = x-1
du = dx
the new integral then equals
2$$\int \frac{1}{\sqrt{1-u^2}} du$$
which in turn equals 2sin^-1(x-1) ??

am I correct?

Thanks again

6. Dec 1, 2003

Ambitwistor

Re: Re: integrals

Yes, you're correct.

7. Dec 1, 2003

noboost4you

Great! Only if I would have known that last week ;)

Thanks again