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How To Integrate By Parts

Integration by parts is a direct consequence of the product rule for the derivative and teh fundamental theorem of calculus. Recall that the product rule says that \[(fg)'=f'g+fg'\]Solving for $fg'$ we have \[fg'=(fg)'-f'g\] By integrating both sides we have \[\int f(x)g'(x)\,dx=f(x)g(x)-\int g(d) f'(x)\,dx\] Denoting $u:=f(x)$ and $dv:=g'(x)\,dx$ , the above integral is more popular in the form \[\int u\,dv=uv-\int v\,du\]

How To Integrate By Substitution

The substitution method based on the fact that under a certain nice condition about $f(x)$ and $g(x)$ we have that \[\int f(g(x))g'(x)dx=\int f(u) du\] where $u=g(x)$ . From the statement above, in many cases it is clear how to choose the $u$ , namely the inner function of the composition function. On the videos below we will discuss how to do integration by substitution. In particular we discuss how to select the right $u$ in many situations. On the third video we discuss two challenging problems that we solve by substitution method.

How to Compute Area

Here we discuss how to compute area using integral. We discuss when we should integrate with respect to a certain variable (y or x) by looking how the boundary of the area looks like. Should you have any question about how to compute area please go to the Ask&watch tab and ask your question there