# 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).

### Exercises

Integrate each of the following functions:

1. int((cos^(-1)2x)^4)/sqrt(1-4x^2)dx

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

3. int(e^x+e^(-x))^(1/4)(e^x-e^(-x))dx

4. int_(pi//3)^(pi//2)(sin\ theta\ d theta)/(sqrt(1+cos\ theta)

5. Find the equation of the curve for which (dy)/(dx)=(1+tan\ 2x)^2sec^2 2x if the curve passes through (2, 1).

6. A space vehicle is launched vertically from the ground such that its velocity v (in km/s) is given by

v=[ln^2(t^3+1)](t^2)/(t^3+1)

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:

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