Problem with these types of integrals

[Jacob87411]

For some reason I just have a problem with these types of integrals..if someone could show me how to do one it would save me a lot of headaches..

1) x+2/(x^2-4) from 0 to 1

This just simplifies to 1/(x-2) so this should be ln(1-2)-ln(0-2)=ln(-1)-ln(-2)?

2) x/x+1, x/x^2+2...Do you use by parts? I cannot get any of these right where its of this nature..thanks so much

 

[EbolaPox]

For number one, you integrated correctly. However, remember your logarithm rules.

$$ln(a) - ln(b) = ln(a/b)$$

Here's a hint for two: If you have something such as$$\int \frac{x dx}{x+a} = \int \frac{x+a-a dx}{x+a} = \int 1dx - \int \frac{a dx}{x+a}$$

You can generalize that further if you have the case x+ k on top. I think that may help.

Furthermore, on the x/x^2 + 2, try u-substitution.

 

[VietDao29]

For some reason I just have a problem with these types of integrals..if someone could show me how to do one it would save me a lot of headaches..

1) x+2/(x^2-4) from 0 to 1

This just simplifies to 1/(x-2) so this should be ln(1-2)-ln(0-2)=ln(-1)-ln(-2)?

This is wrong!
You are forgetting the absolute value. You should note that in the reals, ln(x) is only defined for positive x. Hence, there's no such thing as ln(-1), or ln(-2).
You can look again at the table of integrals, you may find something that reads:
$$\frac{dx}{x} = \ln |x| + C$$

2) x/x+1, x/x^2+2...Do you use by parts? I cannot get any of these right where its of this nature..thanks so much

For the first one, as EbolaPox has pointed out, you should split it into 2 integrals:
$$\int \frac{x}{x + 1} dx = \frac{x + 1 - 1}{x + 1} dx = \int dx - \int \frac{dx}{x + 1}$$.
For the second one, if your numerator is 1 degree less than your denominator, then it's common to try to take the derivative of the denominator, then rearrange the numerator so that it contains a multiple of the derivative of the denominator. Your answer will have a ln part and an arctan part (sometimes it does not have an arctan part).
$$\int \frac{x}{x ^ 2 + 2} dx$$
Now:
(x² + 2)' = 2x. Now try to do some manipulation on the numerator to get a multiple of 2x:
$$\int \frac{x}{x ^ 2 + 2} dx = \int \frac{\frac{1}{2} \times 2x}{x ^ 2 + 2}$$
Now use the u-substitution:
u = x² + 2, we have:
$$\int \frac{x}{x ^ 2 + 2} dx = \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln |u| + C = \frac{1}{2} \ln |x ^ 2 + 2| + C = \frac{1}{2} \ln (x ^ 2 + 2) + C$$.
Since x² + 2 is positive for all x, so:
|x² + 2| = x² + 2.
Can you get it? :)