Recent content by La_Lune

Hi all, Say that I already know W1, W2 are both subspaces of a vector space V, W1∩W2={0}, and that dim(W1)+dim(W2)=dim(V)=n, can I thus conclude that V=W1+W2, namely V is the direct sum of W1 and W2?

I thank you so much~it seems that one should really be flexible while trying to use substitution :)

Thank you so much SammyS! The method for question b worked like a charm! Can you please tell me how I shall proceed with question a? I mean using integration by parts I got this: 2x√(1+e^x)-2∫√(1+e^x)dx but I don't know how to find the integral of the second part...

Homework Statement a) ∫xe^x/√(1+e^x) dx from ln3 to ln8 b)∫arccos(tanx)dx from -π/4 to π/4 Homework Equations uv-∫vdu? Not so sure about this The Attempt at a Solution For question a I tried to make e^x/√(1+e^x)=dv and x=u, but then my result was different from the correct answer, so I guess...