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If $\int_{e}^{x} t f (t) d t = \sin x - x \cos x - \frac{x^{2}}{2}$ for all $x \in R ~ {0}$ then the value of $f (π / 6)$ is

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detailed solution

Correct option is C

Differentiating both the sides and using Property 17, we havexf(x)=cos x – cos x+x sin x – x, so f(x)=sin x–1.Hence f (π/6) = sin π/6 – 1 = – 1/2.

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If ∫ex tf(t)dt=sin⁡x−xcos⁡x−x22 for all x∈R~{0} then the value of f(π/6) is

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If $\int_{e}^{x} t f (t) d t = \sin x - x \cos x - \frac{x^{2}}{2}$ for all $x \in R ~ {0}$ then the value of $f (π / 6)$ is