Page 215 - Calculus Workbook For Dummies

199
Chapter 11: Integration Rules for Calculus Connoisseurs

Q. Find # dx . 4. Determine which trig function is repre-
2
4 x + 25 sented by the radical over the a, then
solve for the radical.
A. 1 ln 4 x + 25 + 2 x + C
2
2 In the figure in Step 2, the radical is on
the Hypotenuse and the a, namely 5, is the
1. Rewrite the function to fit the form H
2
2
u + a . Adjacent side. A is secant so you’ve got
= # dx 4 x + 25
2
2 = seci
2 ^ x + 5 2 5
h
2
4 x + 25 = 5 seci
2. Draw a SohCahToa right triangle
u 2 x
where tani equals , namely . 5. Use the results from Steps 3 and 4
a
5
to make substitutions in the original
Note that when you make the opposite integral and then integrate.
side equal to x2 and the adjacent side
equal to 5, the hypotenuse automatically # dx
2
2
becomes your radical, 4 x + 25. (This 4 x + 25
follows from the Pythagorean Theorem.) 5 sec i di
2
See the following figure. = # 2 ! from Step 3
5 seci ! from Step 4
1
= # sec di
i
2
1 ln sec +i tani + C
=
√4x 2 + 25 2x 2
Get this last integral from your textbook,
the Calculus For Dummies cheat sheet, or
θ from memory.
6. Use Steps 2 and 4 or the triangle to get
5
rid of the seci and tani.
2 x
3. Solve tani = for x, differentiate, 4 x + 25
2
5 = 1 ln + 2 x + C
and solve for dx. 2 5 5
2 x = tani = 1 2 ln 4 x + 25 + 2 x - 1 2 ln5 + C
2
5 1 1
2
5 tani = 2 ln 4 x + 25 + 2 x + C ( 2 ln5 + C is
x =
2 just another
dx = 5 sec i constant , so you
2
di 2 can replace it by C .)
5
2
dx = sec i di
2 Tip: Remember that Step 2 always
u
involves , and Step 4 always involves
a
a . How about U Are Radically
Awesome?