Calculate the Iterated integral

Question

someone helped me already but i wasnt so sure about it, how do i solve it and was 9ln2/2 the good answer? so please help me :)

Answer 1

Hi Finn,

To solve the inner integral (the one with dy), you treat any variable not y (in this problem, just x) as a constant.

The integral of 1/y is ln(y), so the integral of x/y with respect to y is just x*ln(y). Likewise, the integral of y is 1/2 y², so the integral of y/x with respect to y is 1/2 y²/x. Putting this together, the inner integral is

[x*ln(y) + 1/2 y²/x] from y = 1 to y = 2, which evaluates to (x*ln(2) + 2/x) - (x*ln(1) + 1/(2x)) = x*ln(2) + 3/(2x).

Our new integral is

∫⁴₁(x*ln(2) + 3/(2x)) dx = ln(2)/2 x² + 3/2 * ln(x) from 1 to 4 = (8*ln(2) + 3/2*ln(4)) - (ln(2)/2 + 3/2*ln(1)). Using ln(4) = 2*ln(2) and ln(1) = 0, this becomes 21/2 * ln(2).

Doug caught an error in my earlier response, so it has been corrected. If the integrand was "x/y - y/x", the answer would be 9/2 ln(2).

Hope this helps! Feel free to ask questions in the comments if something doesn't make sense.

Akshat Y.