Question

证明定积分等式!f（x）在[1,a^2](a>1)上连续求证：∫x^3f(x^2)dx(上限a下限1)=1/2∫xf(x

Answer 1

证明：∵f(x)在[1,a^2](a>1)上连续∴f(x)在[1,a](a>1)上连续∵原式左边=1/2∫(1,a)x²f(x²)(2xdx)=1/2∫(1,a)x²f(x²)d(x²)=1/2∫(1,a²)xf(x)dx (用x代换x²)=原是右边∴原命题成立...