高等数学求定积分,如下图23题,看了答案有一种方法,不好想,看大神有没有其他的解法.
1个回答

设 x = 2sect,则当 x = -2 时,t = π.当 x = -2√2 时,t = 3π/4.dx = 2(sect*tant)*dt

则原积分变换为：

∫2tant*2(sect*tant)*dt/(2sect)^3 积分限变换为 π 3π/4

=1/2*∫sect*(tant)^2 *dt/(sect)^3

=1/2*∫(tant)^2*(cost)^2 *dt

=1/2*∫(sint)^2*dt

=1/4*∫[2(sint)^2]*dt

=1/4*∫[1-cos2t]*dt

=1/4*∫dt - 1/4*∫cos2t *dt

=1/4*t - 1/8*∫cos2t *d(2t)

=1/4*t - 1/8*sin2t

=1/4*(3π/4 - π) - 1/8*[sin(3π/2) - sin(2π)

=-π/16 - 1/8*(-1 - 0)

=1/8 - π/16

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