Free counters!

Tuesday, October 8, 2013

Integration - Miscellaneous Substitution, 2

Category: Integral Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate


Solution:

Consider the given equation above


You notice that the denominator contains trigonometric functions and we cannot integrate it by simple integration. This is a difficult one because the numerator has no trigonometric functions. If you will use the integration by parts, then the above equation will be more complicated and there will be an endless repetition of the procedure.  

For this type of a function, like the given equation above, we can integrate it by Miscellaneous Substitution. Let's proceed with the integration technique as follows 

Let





From double angle formula,



Since the given problem has Cosine function, then we can get the values of Sine and Cosine functions from Tangent function as follows 

Photo by Math Principles in Everyday Life

Using the figure above that



From the given problem


Substitute the values of dx, Sin x, and Cos x to the above equation, we have








The above equation can now be integrated by Inverse Trigonometric Function Formula as follows



but


Therefore,