Question

`int_(pi/4)^(pi/2)(-e^(cotx))/(sin^2x)dx=`?

I used u-sub for this one, and I got `1/2-1/2e^2`. However, the correct answer is `1-e`.

Is my answer wrong or I need to simplify it? If so, how?

Thank you!

Answer 1

`int_(pi/4)^(pi/2)(-e^(cotx))/(sin^2x)dx=1-e`

Explanation:

For `int_(pi/4)^(pi/2)(-e^(cotx))/(sin^2x)dx`, let `u=cotx`

then `du=-csc^2xdx` and upper and lower limits are `0` and `1` respectively. And

`int_(pi/4)^(pi/2)(-e^(cotx))/(sin^2x)dx`

= `int_1^0e^udu`

= `[e^u]_1^0`

= `1-e`