Question

$$\int \frac{x+\sqrt{x+1}}{x+2}dx$$

Answer 1

$$Wehave,$$
$$I=\int \frac{x+\sqrt{x+1}}{x+2}dx$$
$$\text{Let},x+1=t^2$$
$$\text{Differentiating both sides we get}$$
$$dx=2tdt$$
$$\text{Now, integration becomes}$$
$$I=\int \frac{(t^2-1+t)}{t^2+1}2tdt$$
$$=2\int \frac{t^3+t^2-t}{t^2+1}dt$$
$$=2\int \frac{t^3+t-t+t^2+1-1-t}{t^2+1}dt$$
$$=2\int \frac{t^3+t+t^2+1-t-t-1}{t^2+1}dt$$
$$=2\int \frac{t^3+t}{t^2+1}dt++2\int \frac{t^2+1}{t^2+1}dt+2\int \frac{-2t-1}{t^2+1}dt$$
$$=2\int tdt+2\int dt-2\int \frac{2t}{t^2+1}dt-2\int \frac{1}{t^2+1}dt$$
$$=t^2+\text{2t - 2}\text{log }|t^2+1|-2{\tan }^{-1}t+C$$
$$=(x+1)+2\sqrt{x+1}-2\text{log}|x+2|-2{\tan }^{-1}\sqrt{x+1}+C$$