Question

∫(-√2→√2)(x^5cosxln(1+x²)+√(2-x²))+dx

Answer 1

问：∫(-√2→√2)(x^5cosxln(1+x²)+√(2-x²))+dx

答：(1)∫xcos^2xdx=∫x·(1+cos2x)/2 dx=1/2·(∫xdx + ∫xcos2x dx)=1/4·[x^2 + ∫xd(sin2x)]=1/4·x^2 + 1/4·x·sin2x - 1/4·∫d(sin2x) dx=1/4·x^2 + 1/4·x·sin2x + 1/8·cos2x + C(2)∫ln(1+x^2)dx=x·ln(1+x^2) -∫xd[ln(1+x^2)]=x·ln(1+x^2) -∫2x^2/(1+x^2) dx=x·ln(1+x^2) - 2x + 2arctanx + C(3)∫(2^x+3^x)^2dx=∫(4^x + 2·6^x + 9^x)dx=4^x/ln4 + 2·6^x/ln6 + 9^x/ln9 + C