Post #32

This week in calculus, we went over questions from old APs and had a worksheet on integration, so I will review that.

The integration that I seem to have the most trouble with are all the inverse integrations. The first step in these problems is to recognize it as tan inverse or sin inverse.
The derivative of arcsin is 1/ the square root of 1-u^2
acrsin is -1/ the square root of 1-u^2
acrtan is 1/ 1+ u^2

Example
22. Determine the integral of 1/the square root of 4-t^2 dt on [1,2]
A. pi/6 B. pi/2 C. pi D. pi/4 E. pi/3

This should first be recognized as acrsin.
Next you need to know that the formula of 1/ the square root of a^2 - x^2 is sin -1 (x/a)
Now plug into the formula: sin-1 (t/2)
From there, plug in the bounds and simplify.
sin-1 (1) - sin-1 (1/2)
pi/2 - pi/6 = pi/3
The answer is E.

7. Compute the integral of 4/ 1+4t^2 dt on [0,1/2]
A. 2pi B. pi/2 C. 3pi/2 D. pi E. 0

Factor out the four, so you are left with 4 integral of dt/1+4t^2
This can be recognized and arctan because of the +1 in the bottom.
The 4t^2 at the bottom can also be (2t)^2
Now you can make your u= 2t and your du=2
Since there is no 2 in the problem, you have to multiply by 1/2 to get rid of it.
4(1/2) Integral of dt/ 1+ u^2 on [0,1/2]
Now that we have the formula needed for arctan, all we have to do is integrate then plug in and simplify.
2 tan-1 (u)
2 tan-1(2t) on [0,1/2]
2 ( tan (2)(1/2)) - 2 (tan (2)(0))
2 (pi/4) - 2(0)
pi/2 - 0 = pi/2
The answer is B.

Integration from the worksheet
8. Integrate the (the square root of x - 1) ^2/ the square root of x
u= the square root of x -1 du = 1/2 x^-1/2
since there is a negative exponent, the x^-1/2 will go to the bottom becoming x^1/2, which is what is needed
but there is an extra 2 at the bottom. To get rid of the 2, you will have to multiply by 2
2 integral u^2
2(1/3) u ^3
2/3 u^3
2/3 (the square root of x -1)^3 +C

13. The integral of (1 + 1/t)^3 (1/t^2)
u= 1+ 1/t du= -1/t^2
Multiply the integral by a negative since there is not any in the problem
- integral u^3
- 1/4 u^4
-1/4 (1+ 1/t)^4 +c

Questions:
I can still use help and examples on related rates, area problems, and volume problems.