大一高数求定积分

Question

大一高数求定积分第二题

Answer 1

简单计算一下即可，答案如图所示

Answer 2

需用到公式

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Answer 3

J10=∫(0,π) -xsin^9xd(cosx)

=-xsin^9xcosx|(0,π)+∫(0,π) cosxd(xsin^9x)
=∫(0,π) cosx(sin^9x+9xsin^8xcosx)dx
=∫(0,π) cosxsin^9xdx+9∫(0,π) xsin^8xcos^2xdx
=∫(0,π) sin^9xd(sinx)+9∫(0,π) xsin^8x(1-sin^2x)dx
=(1/10)*sin^10x|(0,π)+9∫(0,π) xsin^8xdx-9∫(0,π) xsin^10xdx
=9*J8-9*J10
J10=(9/10)*J8
同理，可得J8=(7/8)*J6，J6=(5/6)*J4，J4=(3/4)*J2，J2=(1/2)*J0
所以J10=(9/10)*(7/8)*(5/6)*(3/4)*(1/2)*J0
=(9/10)*(7/8)*(5/6)*(3/4)*(1/2)*∫(0,π) xdx
=(9/10)*(7/8)*(5/6)*(3/4)*(1/2)*(x^2)/2|(0,π)
=(9/10)*(7/8)*(5/6)*(3/4)*(1/2)*(π^2)/4