Question

急！！！高数作业，有关定积分，导数，和极限，要详细过程，好的有加分“---（）”为标准答案，要详细过程

1.求导数
①y=∫(x^2,5)sint/t dt -----(-2sin(x)^2/x)
②y=∫(2x,x^2)根号下1+t^3 dt-----(2x*根号下1+x^6-2*根号下1+8x^2)
2.求由参数表示式x=∫(0,t)sinudu,y=∫(0,t)cosudu所给定的函数y对x的导数dy/dx
-----(cott)
3.求由∫(0,y)e^t dt+∫(0,x)cost dt=0所确定的隐函数y对x的导数dy/dx-----(-cosx/e^y)
4.求定积分：①∫(4,9)根号x*(1+根号x)dx----(45又1/6)
②∫(0,根号3)1/a^2+x^2 dx-----(π/3a)
③∫(-e-1,-2)1/1+x dx-----(-1)
④∫(0,2π)|sinx|dx----(4)
⑤设f(x)=x+1 当x<=1;1/2x^2 当x>1 求∫(0,2)f(x)dx-----(8/3)
5.①limx→1 ∫(1,x)e^t^2 dt/lnx-----(e)
②limx→0 ∫(0,x^2)t^3/2 dt/∫(0,x)t*(t-sint)dt-----(12)
就是像题目里写的那样啦，有求导数的还有定积分，极限的题目，我也知道数量是有些多啦，如果实在不行的话，就能写几道写几道好了，标清题号就可以了，麻烦大家了啊
╭(╯3╰)╮

Answer 1

1。
① y = ∫(5->x²) sint / t dt
dy/dx = sin(x²)/(x²) * d(x²)/dx = sin(x²)/(x²) * (2x)
= 2sin(x²) / x
② y = ∫(2x->x²) √(1+t³) dt
dy/dx = √(1+(x²)³) * d(x²)/dx - √(1+(2x)³) * d(2x)/dx
= √(1+x^6) * 2x - √(1+8x³) * 2
= 2x√(1+x^6) - 2√(1+8x³)

2。
x = ∫(0->t) sinu du，dx/dt = sint dt
y = ∫(0->t) cosu du，dy/dt = cost dt
dy/dx = (dy/dt)/(dx/dt) = (cost)/(sint) = cot(t)

3。
∫(0->y) e^t dt + ∫(0->x) cost dt = 0
e^y*dy/dx + cosx*1 = 0
e^y*dy/dx = -cosx
dy/dx = -cosx/e^y

4。
① ∫(4->9) √x*(1+√x) dx
= ∫(4->9) (√x + x) dx
= (2/3)x^(3/2) + x²/2 |(4->9)
= [(2/3)(9)^(3/2) + 9²/2] - [(2/3)(4)^(3/2) + 4²/2]
= 271/6
= 45 + 1/6
② ∫(0->√3) 1/(a² + x²) dx
令x = a*tanz，dx = a*sec²z dz
x = 0，z = 0，x = √3，z = arctan(√3/a)
=> ∫(0->arctan(√3/a)) 1/(a*sec²z) * a*sec²z
= z |(0->arctan(√3/a)
= arctan(√3/a)
③ ∫(-e-1->-2) 1/(1+x) dx
= (-e-1->-2) 1/(1+x) d(1+x)
= ln|1+x| |(-e-1->-2)
= ln(1-2) - ln(1-e-1)
= ln(-1) - ln(-e)
= ln[(-1)/(-e)]
= ln(1/e)
= -1
④ ∫(0->2π) |sinx| dx
= 2∫(0->π) sinx dx
= 2(-cosx) |(0->π)
= -2(-1-1)
= 4
⑤ f(x) = { x+1，x≤1
{ (1/2)x²，x>1
∫(0->2) f(x) dx
= ∫(0->1) (x+1) dx + ∫(1->2) (1/2)x² dx
= 3/2 + 7/6
= 8/3

5。
① lim(x->1) [∫(1->x) e^(t²) dt] / lnx
= lim(x->1) e^(x²) / (1/x) <= 洛必达法则
= e^(1²) / (1)
= e
② lim(x->0) [∫(0->x²) t^(3/2) dt] / [∫(0->x) t(t-sint) dt]
= lim(x->0) [(x²)^(3/2) * d(x²)/dx] / [x(x-sinx) * dx/dx] <= 洛必达法则
= lim(x->0) (x³ * 2x)/(x(x-sinx))
= 2lim(x->0) x³/(x-sinx)
= 2lim(x->0) 3x²/(1-cosx) <= 洛必达法则
= 6lim(x->0) 2x/(sinx) <= 洛必达法则
= 12 * lim(x->0) x/sinx
= 12 * 1
= 12

Answer 2

题意不清，请说明

Answer 3

题目太多