[数据结构与算法]定积分换元法

定理

${\int }_a^b$ f(x) dx = ${\int }_c^d$f [g(t)] g’(t) dt

引入换元函数，上下限也要跟着换

结论

1.f（x）在[-a,a] 上连续且为偶函数，则 ${\int }_{?a}^a$ f(x) dx = 2${\int }_0^a$ f(x) dx

2.f（x）在[-a,a] 上连续且为奇函数，则 ${\int }_{?a}^a$ f(x) dx = 0

3.${\int }_0^{pi/2}$ f(sin x) dx = ${\int }_0^{pi/2}$ f(cos x) dx

4.${\int }_0^{pi}$ x f(sin x) dx = (pi/2)${\int }_0^{pi}$ f(sin x) dx