Evaluate the following integral: `int_0^(pi//2)(sin x cos x)/(cos^2x+3cos x+2)dx`

Explanation

Let`I=int_0^(pi//2)(sin x cos x)/(cos^2x+3cos x+2)dx`

Put `cos=t` then `sinxdx=-dt`

`I_1=-int(tdt)/(t^2+3t+2)`

`I_1=-int(tdt)/((t+1)(t+2))`

`I_1=-int(2dt)/(t+1)+int(dt)/(t+2)`

`I_1=-[2log(t+2)-log(t+1)]`

`I=-[2log(t+2)-log(t+1)]_1^0`

`I=-3log2+2log3`

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