A simple multiple integral!

Calculus Level 3

$\large \int_{0}^{\frac{\pi}{2}} \int_{y}^{\frac{\pi}{2}} \frac{\sin x}{x} dx\ dy = ?$

3 2 1 4

1 solution

Chew-Seong Cheong
Apr 3, 2019

$\begin{aligned} I & = \int_0^\frac \pi 2 \int_y^\frac \pi 2 \frac {\sin x}x \ dx \ dy & \small \color{#3D99F6} \text{Sine integral }\text{Si }(z) = \int_0^z \frac {\sin t}t \ dt \\ & = \int_0^\frac \pi 2 \left(\text{Si }\left(\frac \pi 2\right) - \text{Si }(y) \right) \ dy \\ & = \frac \pi 2 \text{Si }\left(\frac \pi 2\right) - \color{#3D99F6} \int_0^\frac \pi 2 \text{Si }(y) \ dy & \small \color{#3D99F6} \text{By integration by parts} \\ & = \frac \pi 2 \text{Si }\left(\frac \pi 2\right) - \color{#3D99F6} y \text{ Si }(y) \ \bigg|_0^\frac \pi 2 + \int_0^\frac \pi 2 \sin y \ dy \\ & = \frac \pi 2 \text{Si }\left(\frac \pi 2\right) - \frac \pi 2 \text{Si }\left(\frac \pi 2\right) - \cos y \ \bigg|_0^\frac \pi 2 \\ & = \boxed 1 \end{aligned}$

Sir, there is a more elementary way.........We can see the region of integration can be written in another way...........We can write it as $x$ ranging from $0$ to $\frac{\pi}{2}$ and $y$ ranging from $x$ to $\frac{\pi}{2}$ and then integrating first with respect to $y$ first.......

Aaghaz Mahajan - 2 years, 2 months ago

Yes @Aaghaz Mahajan , that's what I had in mind while framing this problem.

Richeek Das - 2 years, 2 months ago

Ohhh I see.......!! Well, this is exactly how most of the questions can be solved without resorting to advanced functions........

Aaghaz Mahajan - 2 years, 2 months ago

I actually suspected it should be solved by switching the order of integration but I thought it would still be obstacles in solving it. You can show the solution and I will edit your LaTex

Chew-Seong Cheong - 2 years, 2 months ago

A simple multiple integral!

1 solution

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