∫[(sinx)^3-(sinx)^5]dx∫x^3(1-x^2)^1/2dx设x=sint,(1-x^2)^1/2=costdx=costdt原式∫x^3(1-x^2)^1/2dx=∫(sint)^3contcostdt=∫(sint)^3(cont)^2dt这步之后.不确定=∫(sint)^3[1-(sint)^2]dt=∫[(sint)^3-(sint)^5]dt=-1/4(cost)^4+1/6(

Question

∫[(sinx)^3-(sinx)^5]dx
∫x^3(1-x^2)^1/2 dx
设x=sint,
(1-x^2)^1/2=cost
dx=cost dt
原式∫x^3(1-x^2)^1/2 dx
=∫(sint)^3 cont cost dt
=∫(sint)^3 (cont)^2 dt
这步之后.不确定
=∫(sint)^3 [1-(sint)^2] dt
=∫[(sint)^3-(sint)^5] dt
= -1/4(cost)^4+1/6(cost)^6+C (这个是我做的答案)
但是正确答案上是
= -1/3(cost)^3+1/5(cost)^5+C

Answer 1

∫(sint)^3 (cost)^2 dt =-∫(sint)^2 (cost)^2 dcost =-∫[1-(cost)^2](cost)^2 dcost =-∫[(cost)^2-(cost)^4]dcost =-1/3(cost)^3+1/5(cost)^5+C