`int_(pi//4)^(pi//2) e^(x)(log sinx+cotx)` का मान कीजिएः

Explanation

`inte^(x)(log sin x+cot x)dx`
`inte^(x) log sin x dx+int e^(x) cotxdx`
`=e^(xd) logsinx-int(cos x)/(sin x).e^(x)dx+inte^(x)cot xdx`
`=e^(xd) logsinx-intcotx.e^(x)dx +inte^(x)cot xdx`
`=e^(x)logsin x`
`therefore underset(pi//4)overset(pi//2)int e^(x)(log sin x+cot x)dx`
`=[e^(x)log sin x]_(pi//4)^(pi//2)`
`=[e^(pi//2)log sin ""(pi)/(2)-e^(pi//4)log sin ""pi/4]`
`=e^(pi//2).log 1-e^(pi//4)log ""1/sqrt2`
`=e^(pi//4).log sqrt2`