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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2015 · Mathematics · Indefinite Integration

The integral ∫ (d x)/(a x+b x) is of the form (1)/(r) In [...

The integral ∫dxacos⁡x+bsin⁡x\int \frac{d x}{a \cos x+b \sin x} is of the form 1rIn⁡\frac{1}{r} \operatorname{In}\left[\tan \left(\frac{x+\alpha}{2}\right)\right]. What is \alpha equal to?

Options

  1. Atan⁡−1(ab)\tan ^{-1} \left(\frac{a}{b}\right)
  2. Btan⁡−1(ba)\tan ^{-1} \left(\frac{b}{a}\right)
  3. Ctan⁡−1(a+ba−b)\tan ^{-1} \left(\frac{a+b}{a-b}\right)
  4. Dtan⁡−1(a−ba+b)\tan ^{-1} \left(\frac{a-b}{a+b}\right)

Correct answer

A. tan⁡−1(ab)\tan ^{-1} \left(\frac{a}{b}\right)

Step-by-step solution

a=rsin⁡αb=rcos⁡α\begin{aligned} & a=r \sin \alpha \\ & b=r \cos \alpha \end{aligned} Dividing (i) from (ii), ab=tan⁡αα=tan⁡−1(ab).\begin{aligned} & \frac{\mathrm{a}}{\mathrm{ b}}=\tan \alpha \\ & \alpha=\tan ^{-1}\left(\frac{\mathrm{a}}{\mathrm{ b}}\right) . \end{aligned}
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