Q118 / 180 · KCET 2014 · Mathematics

If linear function \( f(x) \) and \( g(x) \) satisfy
\( \int[(3 x-1) \cos x+(1-2 x) \sin x] d x=f(x) \cos x+g(x) \sin x+C \), then

  • A
    \( f(x)=3 x-5 \)
  • B
    \( g(x)=3+x \)
  • C
    \( f(x)=3(x-1) \)
  • Correct answer
    D
    \( g(x)=3(x-1) \)

Solution

Given that $I=\int[(3 x-1) \cos x+(1-2 x) \sin x] d x$ $=f(x) \cos x+g(x) \sin x+c$ $\Rightarrow I=(3 x-1) \sin x-\int \sin x \cdot 3 d x+(1-2 x)(-\cos x)+\int \cos x(-2) d x$ $=(3 x-1) \sin x+3 \cos x-\cos x+2 x \cos x-2 \sin x+c$ $=(3 x-1-2) \sin x+(2+2 x) \cos x+c$ $=3(x-1) \sin x+2(x+1) \cos x+c$ Therefore, $f(x)=2(x+1)$ and $g(x)=3(x-1)$

Hence, the correct answer is option: 'D'.

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