作业帮一道微积分竞赛题,求大神!请注意根号里面是sin-sin²x.还有不带根号解答的就别白费劲了~.
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/13 11:04:39
一道微积分竞赛题,求大神!请注意根号里面是sin-sin²x.还有不带根号解答的就别白费劲了~.
一道微积分竞赛题,求大神!
请注意根号里面是sin-sin²x.还有不带根号解答的就别白费劲了~.
一道微积分竞赛题,求大神!请注意根号里面是sin-sin²x.还有不带根号解答的就别白费劲了~.
∫(π/2→π)√(sinx-sin^2(x))dx
=∫(π/2→0)√(sinx-sin^2(x))d(π-x)
=∫(0→π/2)√(sinx-sin^2(x))dx
所以原式=2∫(0→π/2)√(sinx-sin^2(x))dx
=2∫(0→1)√(t-t^2)/√(1-t^2)*dt (t=arcsinx)
=2∫(0→1)√(t/(1+t))dt
=2∫(0→1)t/√(t^2+t)*dt
=4∫(0→1)t/√((2t+1)^2-1)*dt
令2t+1=1/cosy
则原式=4∫(0→arccos(1/3))(1-cosy)/(2cosy)*1/tany*1/2*siny/cos^2(y)dy
=∫(0→arccos(1/3))(1-cosy)/cos^2(y)*dy
=∫(0→arccos(1/3))sec^(y)dy-∫(0→arccos(1/3))dy/cosy
=tany|(0→arccos(1/3))-ln|secy+tany||(0→arccos(1/3))
=2√2-ln(3+2√2)