Question

∫f(x)dx=X²+C，求∫xf(1-x²)dx

Answer 1

由∫f(x)dx=X²+C得f(x)=2x
则∫xf(1-x²)dx=∫2x(1-x²)dx=∫(2x-2x^3)dx=x²-(x^4)/2+C

Answer 2

f(x)=(x^2+C)'=2x
f(1-x^2)=2-2x^2
∫xf(1-x^2)dx
=∫(2x-2x^3)dx
=∫2xdx-∫2x^3dx
=x^2-1/2*x^4+C