The integral _0^/213+2 x+ x,dx is equal to:

Question

The integral $\displaystyle\int_0^{\pi/2}\frac1{3+2\sin x+\cos x}\,dx$ is equal to:

Accepted Answer

With $t=\tan(x/2)$:

$$I = \int_0^1\frac{2\,dt}{3(1+t^2)+2(2t)+(1-t^2)} = \int_0^1\frac{2\,dt}{4+4t+2t^2} = \int_0^1\frac{dt}{(t+1)^2+1}$$

$$= \big[\tan^{-1}(t+1)\big]_0^1 = \tan^{-1}2-\tan^{-1}1 = \tan^{-1}(2)-\frac\pi4$$

Solutions · 1