Question

定积分从一个区间换到另一个区间？

这个题他的区间是怎么来的，为什么要令x=二分之派-t，还有函数怎么变了。数学小白希望各位大佬解答一下谢谢！！！我会积极采纳

Answer 1

I = 4∫[0, π]cos^4 x dx

= 8∫[0, π/2]cos^4 x dx

x = π/2 - t

dx = -dt

I = 8∫[π/2, 0] cos^4 (π/2 - t) d(π/2 - t)

= 8∫[0, π/2] sin^4 t dt

= 8∫[0, π/2] sin^4 x dx

不定积分的公式

1、∫ a dx = ax + C，a和C都是常数

2、∫ x^a dx = [x^(a + 1)]/(a + 1) + C，其中a为常数且 a ≠ -1

3、∫ 1/x dx = ln|x| + C

4、∫ a^x dx = (1/lna)a^x + C，其中a > 0 且 a ≠ 1

5、∫ e^x dx = e^x + C

6、∫ cosx dx = sinx + C

7、∫ sinx dx = - cosx + C

8、∫ cotx dx = ln|sinx| + C = - ln|cscx| + C

Answer 2

Answer 3

I = 4∫[0, π]cos^4 x dx
= 8∫[0, π/2]cos^4 x dx (by symmetry)
x = π/2 - t
dx = -dt
I = 8∫[π/2, 0] cos^4 (π/2 - t) d(π/2 - t)
= 8∫[0, π/2] sin^4 t dt (co-function property and switching lower limit and upper limit)
= 8∫[0, π/2] sin^4 x dx (change t to x)