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The integral of log(x) is not necessarily straight-forward. Though we can use the fact that d/dx(log(x)) = 1/x to help us. Rather than simply trying to integrate log(x), we can use integration by parts on 1 ...

The key to solving any integral of this form is to use the cosine rule: cos(2x) = cos 2 (x) - sin 2 (x) = 2 cos 2 (x) - 1 = 1 - 2 sin 2 (x) All of these forms are really helpful when solving problems such as...

y=x^ x To find the differential, dy/dx, logs of both sides must be taken: log(y)=log(x^ x ) Then using log rules, the power can be brought down, outside the log expression: log(y) = x log(x) This expre ssion...

If you aren't lucky enough to be told in the question, the first thing we have to do is know when we should you integration by parts. Well, like most things in maths this should become second nature to you w...

As you will be aware, it is not easy to directly integrate terms involving sin^2(x) and cos^2(x), so we use a substitution to turn them into something which we can integrate. From the double angle formula fo...