Question

定积分计算问题？

如图，图中划线部分是怎么得出结果的？请解释一下！

Answer 1

如上图

Answer 2

f(x)=∫(0->x) sint/(π-t) dt
∫(0->π) f(x) dx =?
solution:
f(x)=∫(0->x) sint/(π-t) dt
f'(x) = sinx/(π-x)
∫(0->π) f(x) dx

=[xf(x)]|(0->π) -∫(0->π) xf'(x) dx
=πf(π) -∫(0->π) xsinx/(π-x) dx
=π∫(0->π) sinx/(π-x) dx -∫(0->π) xsinx/(π-x) dx
=∫(0->π) sinx. (π-x)/(π-x) dx
=∫(0->π) sinx dx
=-[cosx]|(0->π)
=2

Answer 3