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UPSC NDA I 2015 Mathematics Solved Paper

For the next two (2) items that follow :

The integral

\int\frac{dx}{a\cos x+b\sin x}

is of the form

\frac{1}{r}\ln\left[\tan\left(\frac{x+\alpha}{2}\right)\right]

[NDA I 2015]

Question : 71 of 120

Marks: +1, -0

What is α equal to?

Solution:

From the solution of the above question, the constant angle can written as

\sin\alpha = \frac{a}{\sqrt{a^2+b^2}}

\cos\alpha = \frac{b}{\sqrt{a^2+b^2}}

\tan\alpha = \frac{a}{b}

\alpha = \tan^{-1}\left(\frac{a}{b}\right)

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