∫(0,π/4) ln(1+tanx) dx Let y = π/4 - x then dy = -dxWhen x = 0,y = π/4,when x = π/4,y = 0J = ∫(0,π/4) ln(1+tanx) dx= ∫(π/4,0) ln[1+tan(π/4-y)] -dy= ∫(0,π/4) ln[1 + (tan(π/4)-tany)/(1+tan(π/4)tany)] dy= ∫(0,π/4) ln[1 + (1-tany)/(1
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![∫(0,π/4) ln(1+tanx) dx Let y = π/4 - x then dy = -dxWhen x = 0,y = π/4,when x = π/4,y = 0J = ∫(0,π/4) ln(1+tanx) dx= ∫(π/4,0) ln[1+tan(π/4-y)] -dy= ∫(0,π/4) ln[1 + (tan(π/4)-tany)/(1+tan(π/4)tany)] dy= ∫(0,π/4) ln[1 + (1-tany)/(1](/uploads/image/z/3928664-56-4.jpg?t=%E2%88%AB%280%2C%CF%80%2F4%29+ln%281%2Btanx%29+dx+Let+y+%3D+%CF%80%2F4+-+x+then+dy+%3D+-dxWhen+x+%3D+0%2Cy+%3D+%CF%80%2F4%2Cwhen+x+%3D+%CF%80%2F4%2Cy+%3D+0J+%3D+%E2%88%AB%280%2C%CF%80%2F4%29+ln%281%2Btanx%29+dx%3D+%E2%88%AB%28%CF%80%2F4%2C0%29+ln%5B1%2Btan%28%CF%80%2F4-y%29%5D+-dy%3D+%E2%88%AB%280%2C%CF%80%2F4%29+ln%5B1+%2B+%28tan%28%CF%80%2F4%29-tany%29%2F%281%2Btan%28%CF%80%2F4%29tany%29%5D+dy%3D+%E2%88%AB%280%2C%CF%80%2F4%29+ln%5B1+%2B+%281-tany%29%2F%281)
∫(0,π/4) ln(1+tanx) dx Let y = π/4 - x then dy = -dxWhen x = 0,y = π/4,when x = π/4,y = 0J = ∫(0,π/4) ln(1+tanx) dx= ∫(π/4,0) ln[1+tan(π/4-y)] -dy= ∫(0,π/4) ln[1 + (tan(π/4)-tany)/(1+tan(π/4)tany)] dy= ∫(0,π/4) ln[1 + (1-tany)/(1
∫(0,π/4) ln(1+tanx) dx
Let y = π/4 - x then dy = -dx
When x = 0,y = π/4,when x = π/4,y = 0
J = ∫(0,π/4) ln(1+tanx) dx
= ∫(π/4,0) ln[1+tan(π/4-y)] -dy
= ∫(0,π/4) ln[1 + (tan(π/4)-tany)/(1+tan(π/4)tany)] dy
= ∫(0,π/4) ln[1 + (1-tany)/(1+tany)] dy
= ∫(0,π/4) ln[(1+tany+1-tany)/(1+tany)] dy
= ∫(0,π/4) [ln(2) - ln(1+tany)] dy /*这一行到
= ln(2) * ∫(0,π/4) dy - J 这一行的转换是为什么!*/
2J = ln(2) * (π/4-0)
J = (π*ln2)/8
难道 ln(1+tanx)dx=ln(1+tany)dx
是的话 为什么相等?
∫(0,π/4) ln(1+tanx) dx Let y = π/4 - x then dy = -dxWhen x = 0,y = π/4,when x = π/4,y = 0J = ∫(0,π/4) ln(1+tanx) dx= ∫(π/4,0) ln[1+tan(π/4-y)] -dy= ∫(0,π/4) ln[1 + (tan(π/4)-tany)/(1+tan(π/4)tany)] dy= ∫(0,π/4) ln[1 + (1-tany)/(1
不是说ln(1+tanx)dx=ln(1+tany)dx这两个一样,这两者不能化等号
而是∫(0,π/4) ln(1+tanx) dx 和对于∫(0,π/4) ln(1+tany) dy
当积分形式一样 而被积函数和对应积分变量一样,对应的积分变量取值一样,那么做出来结果是一样的,因为定积分其实质上是一个数
正如∫(0,1) xdx=1/2 ∫(0,1) ydy=1/2 这两个定积分的结果是一样的