几个高数题.积分..不用完全解答..做的差不多给个思路就可以.积分.X^(1/2) * e^(X^(1/2)) dx积分

几个高数题.积分..
不用完全解答..做的差不多给个思路就可以.
积分.X^(1/2) * e^(X^(1/2)) dx
积分.X * (sinx)^2 * cosx dx
积分.1/(X + X * X^(1/2))
积分.(1 + X^(1/2))^8
每个题后面都有个dx..后面俩题忘写了.

积分.X^(1/2) * e^(X^(1/2)) dx
x^1/2 = t,x = t^2
积分.t * e^t dt^2
= 2积分.t^2 * e^t dt
= 2积分.t^2 de^t
= 2(t^2 e^t - 积分.e^t dt^2)
= 2(t^2 e^t - 2积分.te^t dt)
= 2(t^2 e^t - 2(te^t - 积分.e^t dt))
= 2(t^2 e^t - 2(te^t - e^t))
= 2t^2 e^t - 4te^t + 4e^t
= 2x e^(x^1/2) - 4(x^1/2)e^(x^1/2) + 4e^(x^1/2) + C
cos^3(x)=(cos3x+3cosx)/4
积分.X * (sinx)^2 * cosx dx
= 积分.X * (1 - cos^2 x) * cosx dx
= 积分.xcosx dx - 积分.xcos^3 x dx
= 积分.xcosx dx - (1/4)积分.x(cos3x+3cosx) dx
= 积分.xcosx dx - (1/4)积分.xcos3x dx - (3/4)积分.xcosx dx
= (1/4)积分.xcosx dx - (1/36)积分.3xcos3x d(3x)
= (1/4)积分.xdsinx - (1/36)积分.3xcos3x d(3x)
= (1/4)(xsinx - 积分.sinx dx) - (1/36)积分.3x dsin(3x)
= (1/4)(xsinx + cosx) - (1/36)(3xsin3x-积分.sin(3x)d(3x))
= (1/4)(xsinx + cosx) - (1/36)(3xsin3x+cos(3x)) + C
积分.1/(X + X * X^(1/2)) dx
= 积分.1/(X + X^(3/2)) dx
x^1/2 = r
= 积分.2r/(r^2 + r^3) dr
= 积分.2/(r + r^2) dr
= 2积分.1/(r(r+1)) dr
= 2积分.(r+1-r)/(r(r+1)) dr
= 2(积分.(r+1)/(r(r+1)) dr - 积分.r/(r(r+1)) dr)
= 2(积分.1/r dr - 积分.1/(r+1) d(r+1))
= 2(ln|x^1/2| - ln|x^1/2+1|) + C
= 2ln|x^1/2| - 2ln|x^1/2+1| + C
= 2(ln|r| - ln|r+1|)
积分.(1 + X^(1/2))^8 dx
= 积分.(1 + x^(1/2))^8 d((x^(1/2))^2)
x^1/2 = k
= 积分.2k(1 + k)^8 d(k)
= 积分.2k(1 + k)^8 d(1+k)
= (2/9)积分.k d(1+k)^9
= (2/9)((1+k)^9 k - 积分.(1+k)^9dk)
= (2/9)((1+k)^9 k - 积分.(1+k)^9d(1+k))
= (2/9)((1+k)^9 k - (1/10)(1+k)^10) + C
= (2/9)((1+x^1/2)^9 x^1/2 - (1/10)(1+x^1/2)^10) + C
不一定对哦 我自己做的 = =.