8. Integration by Trigonometric Substitution
by M. Bourne
In this section, we see how to integrate expressions like
`int(dx)/((x^2+9)^(3//2))`
Depending on the function we need to integrate, we substitute one of the following to simplify the expressions to be integrated:
For `sqrt(a^2-x^2)`, use ` x =a\ sin\ theta`
For `sqrt(a^2+x^2)`, use ` x=a\ tan\ theta`
For `sqrt(x^2-a^2)`, use `x=a\ sec\ theta`
After we use these substitutions we'll get an integral that is "do-able".
Example 1
`int(dx)/((x^2+9)^(3//2))`
Example 2
`int_4^5(sqrt(x^2-16))/(x^2)dx`
Exercises
Integrate each of the given functions:
1. `intsqrt(16-x^2)dx`
2. `int(3\ dx)/(xsqrt(4-x^2))`
3. `int(dx)/(sqrt(x^2+2x))`
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