The integral (1-1√3)( x- x)1+2√32x,dx is equal to

Question

The integral $\displaystyle\int\frac{\left(1-\frac1{\sqrt3}\right)(\cos x-\sin x)}{1+\frac2{\sqrt3}\sin2x}\,dx$ is equal to

Accepted Answer

Writing $\cos x-\sin x = \sqrt2\sin\left(\dfrac\pi4-x\right)$ and $1+\dfrac2{\sqrt3}\sin2x = \dfrac2{\sqrt3}\left(\sin\dfrac\pi3+\sin2x\right)$, then using sum-to-product on the denominator, the integrand reduces to a combination of $\csc$ terms:

$$\frac12\int\left[\csc\left(\frac\pi6+x\right)-\sec\left(\frac\pi6-x\right)\right]dx$$

Integrating each standard $\csc,\sec$ form and combining logs gives

$$\frac12\log_e\left|\frac{\tan\left(\frac x2+\frac{\pi}{12}\right)}{\tan\left(\frac x2+\frac\pi6\right)}\right|+C$$

Solutions · 1