微积分...

∫(x^7)[e^（x^2)] dx
=(1/8)∫[e^（x^2)] d(x^8)
设x^2=t,则x^8=t^4
原式=(1/8)∫[e^t] d(t^4)

∫(x^7)(e^x^2) dx
=∫(x^6)/2d(e^x^2)
=x^6e^x^2/2-∫(e^x^2)/2d(x^6)
=x^6e^x^2/2-∫(e^x^2)/2*(6x^5)dx
=x^6e^x^2/2-∫[(3/2)x^4]d(e^x^2)
=x^6e^x^2/2-[(3/2)x^4](e^x^2)+∫e^x^2*(6x^3)dx
=x^6e^x^2/2-[(3/2)x^4](e^x^2)+∫(3x^2)d(e^x^2)
=x^6e^x^2/2-[(3/2)x^4](e^x^2)+(3x^2)*(e^x^2)-3∫(e^x^2)d(x^2)
=x^6e^x^2/2-[(3/2)x^4](e^x^2)+(3x^2)*(e^x^2)-3(e^x^2)(x^2)+C

原式=∫x^7e^(x^2)dx=
(1/2)∫x^6d(e^(x^2))=
(1/2)x^6*e^(x^2)-(1/2)∫e^(x^2)d(x^6)=
(1/2)x^6*e^(x^2)-3∫x^5*e^(x^2)dx
重复利用这个思路最后可求得不定积分，后面的步骤比较多，就先不写了，你再自己算吧