Question

Integration problem with odd functions?

Answer 1

Let f(x) be an odd function with \[\int\limits_{3}^{-2}[5f(x)+2]dx=20\]. Evaluate \[\int\limits_{3}^{2}f(x)dx\]

Answer 2

hmm start with
\[\int_{-2}^35[f(x)+2]dx=-20\] maybe that will help

Answer 3

then this is
\[5\int_{-2}^3f(x)dx+10=-20\] so
\[5\int_{-2}^3f(x)dx=-30\] maybe we can get something out of that

Answer 4

divide by \(5\) and get
\[\int_{-2}^3f(x)dx=-6\]

Answer 5

now we are almost done
can you see how to finish it from there? notice that we have not yet used the fact that \(f\) is odd

Answer 6

hmm, i just know that \[\int\limits_{a}^{-a}f(x)dx = 0\] but im not sure how to apply this rule to this kind of problem :/

Answer 7

the gimmick is to note that
\[\int_{-2}^2f(x)dx=0\]

Answer 8

then since \[\int_{-2}^3f(x)dx=-6=\int_{-2}^2f(x)dx+\int_ 2^3f(x)dx=0+\int_2^3f(x)dx\]

Answer 9

that makes
\[\int_2^3f(x)dx=-6\] and so
\[\int_3^2f(x)dx=6\]

Answer 10

oh shoot. i actually typed the wrong problem. Ill retype it now; my prof said the answer is 2. let f(x) be an odd function with \[\int\limits_{-2}^{3}[5(fx)+2]dx=20\]. Evaluate \[\int\limits_{2}^{3}f(x)dx\]. sorry about hat