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Integration - Miscellaneous Substitution

__Category__: Integral Calculus, Trigonometry
"Published in Newark, California, USA"
Find the integral for

__Solution__:

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 the double angle formula,

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

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

but

Therefore,