Given Definite Integral can be written as:
⇒ I(x) = \(\int\limits_0^\pi\cfrac{1}{3+2sin\text x+cos\,\text x}d\text x \)
We know that:
![](https://www.sarthaks.com/?qa=blob&qa_blobid=10163281322380144776)
![](https://www.sarthaks.com/?qa=blob&qa_blobid=6987648553149919607)
![](https://www.sarthaks.com/?qa=blob&qa_blobid=16597959769684359850)
We know that: 1 + tan2x = sec2x
![](https://www.sarthaks.com/?qa=blob&qa_blobid=14796971863815128085)
Let us assume,
y = 1 + tan\((\cfrac{\text x}2)\)
Differentiating w.r.t x on both sides we get,
![](https://www.sarthaks.com/?qa=blob&qa_blobid=13287821437650539213)
Upper limit for the Definite Integral:
![](https://www.sarthaks.com/?qa=blob&qa_blobid=16705123681853845498)
Lower limit for the Definite Integral:
![](https://www.sarthaks.com/?qa=blob&qa_blobid=17785285135750553102)
Substituting (2),(3),(4) in the eq(1), we get,
![](https://www.sarthaks.com/?qa=blob&qa_blobid=7734857809854318710)
We know that:
![](https://www.sarthaks.com/?qa=blob&qa_blobid=9420893866183789616)
We know that:
![](https://www.sarthaks.com/?qa=blob&qa_blobid=15192149822662489926)
[here f’(x) is derivative of f(x)).
![](https://www.sarthaks.com/?qa=blob&qa_blobid=11954494107800972213)