A calculus problem by Aakash Khandelwal

Calculus Level 4

Suppose $f(x) = \displaystyle\int_{\pi/3}^{x} \dfrac{\sin(t)}{t} \, dt$ .

Now let $\displaystyle\alpha = \dfrac{\pi}{2} f(\pi/2)- \int_{\pi/3}^{\pi/2} f(x)\, dx$ .

Then find the value of $356 \times \alpha$ .

The answer is 178.

1 solution

Brian Moehring
Feb 26, 2017

By using integration by parts and the fundamental theorem of calculus, $\begin{aligned}\int_{\pi/3}^{\pi/2} f(x)\, dx &= \frac{\pi}{2}f(\pi/2) - \frac{\pi}{3}f(\pi/3) - \int_{\pi/3}^{\pi/2} xf'(x)\,dx \\ &= \frac{\pi}{2}f(\pi/2) - 0 - \int_{\pi/3}^{\pi/2} \sin(x)\,dx \\ &= \frac{\pi}{2}f(\pi/2) - \frac{1}{2}.\end{aligned}$

By rearranging, we see that $\alpha = 1/2$ , so that $356\alpha = \boxed{178}$ .

A calculus problem by Aakash Khandelwal

The answer is 178.

1 solution

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