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NDA 1 2024 Maths, question 81

Topic: Integral calculus

Passage · Q81 to Q82

Given that ∫3cos⁡x+4sin⁡x2cos⁡x+5sin⁡xdx=αx29+β29ln⁡∣2cos⁡x+5sin⁡x∣+c\int \frac{3\cos x+4\sin x}{2\cos x+5\sin x} dx=\frac{\alpha x}{29}+\frac{\beta}{29}\ln |2\cos x+5\sin x|+c

Q81

What is the value of α\alpha ?

  1. (a) 77
  2. (b) 1313
  3. (c) 1717
  4. (d) 2626Correct answer

Solution.

Answer: (d) 2626

  1. Put D = 2 cos x + 5 sin x, so D′ = 5 cos x − 2 sin x.
  2. Comparing 29(3 cos x + 4 sin x) with αD + βD′ gives 2α + 5β = 87 and 5α − 2β = 116.
  3. Solving yields α = 26, β = 7.