Question

大学数学经济数学定积分 课后题

大学数学经济数学定积分 课后题求解答3.4.5 麻烦你过程详细谢谢喔

Answer 1

3、∫(0,π) [f(x)+f''(x)]sinxdx
=∫(0,π) f(x)sinxdx+∫(0,π) f''(x)sinxdx
=-∫(0,π) f(x)d(cosx)+∫(0,π) sinxd[f'(x)]
=-f(x)cosx|(0,π)+∫(0,π) cosxf'(x)dx+sinxf'(x)|(0,π)-∫(0,π) f'(x)cosxdx
=[sinxf'(x)-cosxf(x)]|(0,π)
=f(π)+f(0)
=1+f(0)
=3
所以f(0)=2
4、∫(0,2) x^2*f''(x)dx
=∫(0,2) x^2d[f'(x)]
=x^2*f'(x)|(0,2)-∫(0,2) f'(x)*2xdx
=4f'(2)-∫(0,2) 2xd[f(x)]
=-2xf(x)|(0,2)+∫(0,2) 2f(x)dx
=-4f(2)+2*1
=-4*0.5+2
=0
5、∫(a,b) xf''(x)dx
=∫(a,b) xd[f'(x)]
=xf'(x)|(a,b)-∫(a,b) f'(x)dx
=bf'(b)-af'(a)-f(x)|(a,b)
=bf'(b)-af'(a)-f(b)+f(a)
=[bf'(b)-f(b)]-[af'(a)-f(a)]