Question

Can I get a step by step instructions on how I can solve a definite integral? Like this one for example:

Answer 1

\[\int\limits\limits_{-2}^{2}(3y ^{2}-5y ^{4})dy\]
I think I can rewrite it as :
\[2\int\limits\limits_{0}^{2}(3y ^{2}-5y ^{4}dy) \]
what will happen after this?

Answer 2

Oh sorry. I think I can rewrite it like this:
\[2\int\limits\limits\limits_{0}^{2}(3y ^{2}-5y ^{4})dy \]

Answer 3

First, you can do that only because it is a symmetrical function, usually that is not possible, but doing so does not make it easier for you to calculate the integral. The first thing you really need to do is to find a primitive for the function 3y^2-5y^4, wich is the function that when derived will give the function you have. In the case os polynomials it is easy. The derivative is given by:\[\frac{d}{dx}ax^b=bax^{b-1}\]So the function that is the primitive for your problem is:\[y^3-y^5\]To make sure that is correct you can take its derivative:

Answer 4

\[\frac{d}{dx}(y^3-y^5)=3x^2-5y^4\]With the primitive you now only need to calculate the integral:\[2\int\limits_{0}^{2}\frac{d}{dy}(y^3-y^5)dy=2\left[y^3-y^5\right]_{0}^{2}=2(8-32)=-48\]