Question

帮忙做不定积分：∫sinxcos^3(x)e^(1-sin^2(x))dx

Answer 1

I = ∫sin(x)*cos^3(x)*(e^(1-sin^2(x)))*dx

= -∫cos^3(x)*(e^(cos^2(x)))*d(cos(x))

令t=cos(x)：

I = -∫(t^3)*(e^(t^2))dt

= -(1/2)*∫(t^2)*(e^(t^2))*d(t^2)

令s=t^2：

I = -(1/2)*∫s*(e^s)*ds

分部积分,得：

I = -(1/2)*s*(e^s)+(1/2)*(e^s)

将s=t^2=cos^2(x)代入,得：

I = -(1/2)*(cos^2(x))*(e^(cos^2(x)))+(1/2)*(e^(cos^2(x)))