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NDA 2019 Maths · Mathematics · Integrals · easy

Using the substitution u=ln x, the integral ∫(cos (ln x))/(x) dx is evaluated as

Single correct option

Using the substitution u=ln⁡xu=\ln x, the integral ∫cos⁡(ln⁡x)x dx\displaystyle\int\frac{\cos(\ln x)}{x}\,dx is evaluated as
  1. A. sin⁡(ln⁡x)+C\sin(\ln x)+C
  2. B. cos⁡(ln⁡x)+C\cos(\ln x)+C
  3. C. xsin⁡(ln⁡x)+Cx\sin(\ln x)+C
  4. D. −sin⁡(ln⁡x)+C-\sin(\ln x)+C
Show answer and solution

Answer: (A) sin⁡(ln⁡x)+C\sin(\ln x)+C

With u=ln⁡xu=\ln x we have du=dxxdu=\frac{dx}{x}. The integral becomes ∫cos⁡u du=sin⁡u+C=sin⁡(ln⁡x)+C\int\cos u\,du=\sin u+C=\sin(\ln x)+C. Differentiating the answer gives cos⁡(ln⁡x)⋅1x\cos(\ln x)\cdot\frac1x, which confirms it. The extra factor xx in one distractor would appear only if dxdx had no denominator.
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Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake