Page 34 - Calculus for the Clueless

Another group of integrals are related to the last group. They can be very involved, but we will do two
moderate ones.

Example 19—

2
2
2
We do this by completing the square: 3x - 18x + 75 = 3(x - 6x + 25) = 3(x - 6x + 9 + 16)= 3[(x
2
- 3) + 16].

u = x - 3; x = u + 3; 2x - 3 = 2(u + 3) - 3 = 2u + 3; du = dx.

Now split the integral.

Both of these integrals should be
known by sight!

2
since u = x - 3 and u + 16 = (x- 3) + 16 = x - 6x + 9 + 16 = x - 6x + 25.
2
2
2
Example 20—
2
2
2
Again we complete the square: 15 + 2x - x = - 1(x - 2x) + 15 = -1(x - 2x + 1) + 16 = 16 - (x -
2
1) .

Again you should be able to tell this is an arc sin; that is, where a = 4 and u = x
- 1.