Question

高中定积分

Answer 1

追答

利用图形，
把积分转化为阴影部分的面积

Answer 2

Answer 3

设 x = sinα。则 dx = cosα*dα 。α 的积分范围变换为：[π/6, π/2]
那么，上面的积分变换为：
=∫cosα * (cosα * dα)
=∫cos²α * dα
=1/2*∫(2cos²α)*dα
=1/2*∫(1+cos2α)*dα
=1/2*∫dα + 1/2*∫cos2α*dα
=1/2*α + 1/4*∫cos2α*(2*dα)
=1/2*α + 1/4*∫cos2α*d(2α)
=1/2*α + 1/4*sin(2α)
=1/2*(π/2 - π/6) + 1/4* [sin(2*π/2) - sin(2*π/6)]
=π/6 + 1/4 * [sinπ - sin(π/3)]
=π/6 - 1/4 * √3/2
=π/6 - √3/8

Answer 4