The value of _/3^/2(2+3 x)/( x(1+ x)),dx is equal to:

Question

The value of $\displaystyle\int_{\pi/3}^{\pi/2}\frac{2+3\sin x}{\sin x(1+\cos x)}\,dx$ is equal to:

Accepted Answer

Split as $2\displaystyle\int\frac{dx}{\sin x(1+\cos x)} + 3\int\frac{dx}{1+\cos x}$.

For the second: $\displaystyle\int_{\pi/3}^{\pi/2}\frac{dx}{1+\cos x} = \int(\csc^2(x/2)-\cot(x/2)\csc(x/2))\,dx$-type manipulation gives $1-\dfrac1{\sqrt3}$ over the limits.

For the first, with $t=\tan(x/2)$: $\displaystyle\int\frac{1+t^2}{2t}\,dt$-type reduction gives $\dfrac16+\dfrac12\ln\sqrt3$ over the limits.

Combining:

$$I = 2\left(\frac16+\frac12\ln\sqrt3\right) + 3\left(1-\frac1{\sqrt3}\right) = \frac{10}3+\ln\sqrt3-\sqrt3$$

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