Evaluate the following Integrals:

Given Definite Integral can be written as:


……(1)


We know that:


And


Let us find the value of 5cosx+3sinx




We know that: 1+tan2x = sec2x


……(2)


Substituting (2) in (1) we get,



Let us assume:


Differentiating on both sides w.r.t x we get,



……(3)


The upper limit for the Definite Integral:



t=1……(4)


The lower limit for the Definite Integral:



t=0……(5)


Substituting (3),(4),(5) in the eq(1) we get,




We need to convert the denominator into standard forms





We know that:


In this problem the values,


Using these values and the standard result, we get,






We know that:


[here f’(x) is derivative of f(x))




We know that:








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