Definite integral problem

Catsmokinpot

When integrating a series containing multiple terms, I ran into a problem.

the problem is that part of the integral required substitution, do i then change the limits for the entire integral, or just part of the integral?

Integral from 0 to π of [4π(1/4 + sin(x) + (sin(x))^2) - π] dx

the (sin(x))^2 is the term i used substitution with.

I rewrote it as 1/2(1-cos(2x)) and substituted U = 2x

click_here!!!

I'd guess part of the integral.

According to Wolfram Alpha, the correct answer is roughly 44.872. Check which way gives you this answer, then you know the right way to do it.

See Wolfram Alpha link:

http://www.wolframalpha.com/input/?i=Integral+from+0+to+pi+of+%284*pi*%281%2F4+%2B+sin%28x%29+%2B+%28sin%28x%29%29^2%29+-+pi%29+dx

sponsoredwalk

If the integral is

[latex] \int_0^{ \pi} \ [ \ 4 \pi ( \ \frac{1}{4} \ + \ \sin(x) \ + \ \sin^2(x) \ ) \ - \ \pi \ ] \ \ dx[/latex]

you can cancel some terms & you'll end up with an integral of the form

[latex] \int_0^{ \pi} \ [ \lambda_1 \cdot f(x) \ + \ \lambda_2 \cdot g(x) \ ] \ dx[/latex]

where the λ's are constants, f(x) &g(x) are just your sine functions.

Because

[latex] \int_0^{ \pi} \ [ \lambda_1 \cdot f(x) \ + \ \lambda_2 \cdot g(x) \ ] \ dx \ = \ \int_0^{ \pi} \ ( \lambda_1 \cdot f(x) \ ) \ dx \ + \ \int_0^{ \pi} \ ( \ \lambda_2 \cdot g(x) \ ) \ dx[/latex]

you can then use u-substitution on one part of the integral, i.e. the
part dealing with f(x) here (for example!), & you don't have to change
anything dealing with g(x).

Fringe

The integral is linear so that means you can integrate each term separately. ie,

Int_a^b ( f(x) + g(x) ) dx = Int_a^b f(x) dx + Int_a^b g(x) dx

You can use a change of variables for just one integral, ie
= Int_a^b f(x) dx + Int_c^d g(u) du

So you can do everything separately but just make sure which variable your are integrating with respect to.

MathsManiac

The other option of course is that you can leave your limits unchanged, and, after doing the (indefinite) integral, substitute the original variable back in.

Then you can evaluate all terms in the definite integral using the original limits.

Another useful thing you can do is that, if you're switching variables in a question like this, you can clarify what variable the limits apply to by writing, for example, x=0 and x=Pi at the bottom and top of the integral sign instead of just 0 and Pi.