Question

Evaluate: $\int \frac{\sin x-\cos x}{e^x+\sin x}dx$

Answer 1

$\int \frac{\sin x-\cos x}{e^x+\sin x}dx$

$=\int \left(\frac{\sin x+e^x-e^x-\cos x}{e^x+\sin x}\right)dx$

$=\int (\frac{\sin x+e^x}{e^x+\sin x}-\frac{(e^x+\cos x)}{e^x+\sin x})dx$

$=\int dx-\int \frac{e^x+\cos x}{e^x+\sin x}dx$

Let $e^x+\sin x=t$

$(e^x+\cos x)dx=dt$

$=x-\int \frac{dt}{t}$

$=x-lnt+c$

$=x-ln(e^x+\sin x)+c$

$\therefore \int \left(\frac{\sin x-\cos x}{e^x+\sin x}\right)dx=x-ln(e^x+\sin x)+c$.