1. Integration: The General Power Formula
by M. Bourne
In this section, we apply the following formula to trigonometric, logarithmic and exponential functions:
`intu^ndu=(u^(n+1))/(n+1)+C\ \ \ (n!=-1)`
(We met this substitution formula in an earlier chapter: General Power Formula for Integration.)
Example 1: Integrate: `intsin^(1//3)\ x cos x dx`
Example 2: Integrate: `int(sin^(-1)4x)/sqrt(1-16x^2)dx`
Example 3: Integrate: `int((3+ln\ 2x)^3)/xdx`
Example 4: Integrate: `int2sqrt(1-e^(-x))e^(-x)dx`
Example 5: Find the equation of the curve for which `(dy)/(dx)=((ln\ x)^2)/x` if the curve passes through `(1, 2)`.
Graph of the solution equation for Example 5, passing through (1, 2).
Integrate each of the following functions:
`int_1^e((1-2 ln x))/xdx`
The shaded region represents the integral in Exercise 2.
`int_(pi//3)^(pi//2)(sin\ theta\ d theta)/(sqrt(1+cos\ theta)`
The shaded region represents the integral we needed to find in Exercise 4.
Find the equation of the curve for which `(dy)/(dx)=(1+tan\ 2x)^2sec^2 2x` if the curve passes through `(2, 1)`.
Graph of the solution `y=(1+tan 2x)^3/6-0.675`, passing through (2, 1).
A space vehicle is launched vertically from the ground such that its velocity v (in km/s) is given by
where t is the time in seconds. Find the altitude of the vehicle after 10.0 s.
The graph of `v=[ln^2(t^3+1)](t^2)/(t^3+1)` is as follows:
The shaded region represents the altitude we need to find in Exercise 6.