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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:

Exercise 1


Exercise 2

`int_1^e((1-2 ln x))/xdx`


The shaded region represents the integral in Exercise 2.

Exercise 3


Exercise 4

`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.

Exercise 5

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).

Exercise 6

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.

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