Question

integral of |3^x-2^x|dx?

Answer 1

as x goes from -1 to 1?
sorry i forgot to put that in the question.

Answer 2

lets compare 3^x and 2^x
for 3^x > 2^x
x ln 3 > x ln 2
x(ln 3 - ln 2)>0
thus for x>0, 3^x > 2^x or 3^x - 2^x is positive
for x=0 , 3^x -2^x =0
for x<0, 3^x - 2^x is -ve
so our inetgral can be broken like this :
integral(|3^x-2^x|) [-1,1] = integral(2^x - 3^x) [-1,0) + 0 + integral ( 3^x -2^x) (0,1]

can you simplify further?

Answer 3

\[f(x) = \left(\begin{matrix}-(3^x - 2^x), for (3^x - 2^x) < 0 \\ (3^x - 2^x), for(3^x-2^x) \ge 0\end{matrix}\right)\]