Browse Questions

# Evaluate the following $\large\int\frac{(1+\cos x)}{x+\sin x}dx$

Toolbox:
• If $f(x)$ is substituted by t, then $f'(x)dx=f'(t)dt.$ Hence $\int f(x)dx=\int t.dt$
• $\large\int \frac{dx}{x}=log |x|+c$
Let $I=\large\int\frac{(1+\cos x)}{x+\sin x}dx$
Let $x+\sin x=t$ Differentiating on both sides.
$(1+\cos x)dx=dt$ on substituting,
I can be written as
$I=\large\int \frac{dt}{t}$
On integrating we get
$I=log\; t+c$
Substituting for t we get
$I=log|x+\sin x|+c$