Evaluate the indefinite integral. (use c for the constant of integration.) sin(2x) 24 + cos2(x) dx

Possibly you refer to the integral,[tex]\displaystyle\int\frac{\sin2x}{24+\cos^2x}\,\mathrm dx[/tex]Recall the double angle identity,[tex]\cos^2x=\dfrac{1+\cos2x}2[/tex]Then the integral is equivalent to[tex]\displaystyle\int\frac{\sin2x}{24+\frac{1+\cos2x}2}\,\mathrm dx=2\int\frac{\sin2x}{49+\cos2x}\,\mathrm dx[/tex]Let [tex]y=49+\cos2x[/tex] so that [tex]\mathrm dy=-2\sin2x\,\mathrm dx[/tex]. Then the integral becomes[tex]\displaystyle-\int\frac{\mathrm dy}y=-\ln|y|+C[/tex][tex]\implies\displaystyle\int\frac{\sin2x}{24+\cos^2x}\,\mathrm dx=-\ln|49+\cos2x|+C[/tex]We can rewrite this to try to get it in a form that more resembles the original integrand.[tex]-\ln|49+\cos2x|+C=-\ln\left|2\left(24+\dfrac{1+\cos2x}2\right)\right|+C[/tex][tex]=-\ln2-\ln\left|24+\cos^2x\right|+C[/tex][tex]=-\ln\left|24+\cos^2x\right|+C[/tex]where the constant term [tex]-\ln2[/tex] got absorbed into the general constant [tex]C[/tex].