Find $\int\frac{e^{2x}-e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)}dx$.

Question

Find $$\int\frac{e^{2x}-e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)}dx$$ I observed that $(e^x\sin x+\cos x)'=e^x\cos x+e^x\sin x-\sin x$ and I tried to denote this by $t$ to change the variable but I failed. Any help?

Answer 1

let $f(x)=e^x\cos x-\sin x$ and

$g(x)=e^x\sin x+\cos x$

and $e^{2x}-e^x+1=(e^x\cos x-\sin x)(e^x\sin x+\cos x)'-(e^x\sin x+\cos x)(e^x\cos x-\sin x)'$

integration is $\displaystyle \int\frac{f(x)g'(x)-g(x)f'(x)}{f(x)g(x)}dx$

$$\int\frac{g'(x)}{g(x)}dx-\int\frac{f'(x)}{f(x)}dx=\ln\bigg|\frac{g(x)}{f(x)}\bigg|+C$$