First slide

Evaluation of definite integrals

Question

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

Easy

A

0
B

1
C

– 1/2
D

3/2

Solution

Differentiating both the sides and using Property 17, we have
$x f (x) = \cos x - \cos x + x \sin x - x, s o f (x) = \sin x - 1 .$
Hence $f (π / 6) = \sin π / 6 - 1 = - 1 / 2 .$

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