$\displaystyle \int \left(3^x+\dfrac{\sin x - \cos x}{5}\right)\,\mathrm{d}x$

$\displaystyle \int \left(3^x+\dfrac{\sin x - \cos x}{5}\right)\,\mathrm{d}x= \int 3^x\,\mathrm{d}x+\dfrac{1}{5}\int \sin x\,\mathrm{d}x-\dfrac{1}{5}\int \cos x\,\mathrm{d}x= \bbox[#FFECB3,5px]{\dfrac{3^x}{\ln 3}-\dfrac{1}{5}\cos x-\dfrac{1}{5}\sin x+k}$

$\displaystyle \int \tan^2 x\,\mathrm{d}x$

$\displaystyle \int \tan^2 x\,\mathrm{d}x= \int \left[(1+\tan^2 x)-1\right]\,\mathrm{d}x= \bbox[#FFECB3,5px]{\tan x - x+k}$

$\displaystyle \int \dfrac{2^{2x+3}}{3^{x-1}}\,\mathrm{d}x$

$\displaystyle \int \dfrac{2^{2x+3}}{3^{x-1}}\,\mathrm{d}x= \int \dfrac{2^3 2^{2x}}{3^{-1}3^{x}}\,\mathrm{d}x= \dfrac{8}{3}\int \dfrac{\left(2^2\right)^{x}}{3^{x}}\,\mathrm{d}x= \dfrac{8}{3}\int \left(\dfrac{4}{3}\right)^x\,\mathrm{d}x= \bbox[#FFECB3,5px]{\dfrac{8}{3\ln(4/3)} \left(\dfrac{4}{3}\right)^x +k}$

$\displaystyle \int \left(2^{2x}+\dfrac{4}{1+x^2}-\sin x\right)\,\mathrm{d}x$

$\displaystyle \int \left(2^{2x}+\dfrac{4}{1+x^2}-\sin x\right)\,\mathrm{d}x= \int 4^x\,\mathrm{d}x +4\int \dfrac{1}{1+x^2}\,\mathrm{d}x -\int \sin x\,\mathrm{d}x= \bbox[#FFECB3,5px]{\dfrac{4^x}{\ln 4}-4\tan^{-1} x+\cos x+k}$

$\displaystyle \int \dfrac{8x^2-3x+4}{x+1}\,\mathrm{d}x$

$\displaystyle \int \dfrac{8x^2-3x+4}{x+1}\,\mathrm{d}x= \int \dfrac{(8x-11)(x+1)+15}{x+1}\,\mathrm{d}x= \int (8x-11)\,\mathrm{d}x+\int \dfrac{15}{x+1}\,\mathrm{d}x= \bbox[#FFECB3,5px]{4x^2-11x+15\ln|x+1|+k}$