using integral by part on this integral

Question

anonymous · Answer

attachment

lgbasallote · Answer

this is double integral?

lgbasallote · Answer

$$\large \int_1^e (\int _0^x \frac{\sin t}{t} dt) dx?$$

anonymous · Answer

yes

anonymous · Answer

but i think the inside integral could not be solve

anonymous · Answer

how can i use integral by part to calculate?

anonymous · Answer

u = sin x , v = 1/t

anonymous · Answer

i'm still confused

anonymous · Answer

$$u=\sin t$$ $$du=\cos t dt$$ $$dv=1/t dt$$ $$v=\ln t$$ $$\int\limits_{?}^{?}udv=uv-vdu$$

richyw · Answer

what level of math are you in? There is no basic antiderivative but the inside integral is the definition of the sine integral;

$$\int\limits_{0}^{x}(\sin t/t)dt =Si(x)$$

$$\int\limits\limits_{1}^{e}Si(x)dx=\cos(e)+(e)Si(e)-\cos(1)-Si(1)$$

Is the answer, but I only know this because I was doing some physics that required it. The math is beyond me, it's like 3rd or 4 year material

richyw · Answer

well not the first integral, that you can do with series expansion but I have no idea how to show the second integral with IBP

anonymous · Answer

would you mind just show how can do this by integral by part?