What is the integral of (sin(t))/(1+sin(t)) ?

The integral of (sin(t))/(1+sin(t)) is 2/(tan(t/2)+1)+t+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (sin(t))/(1+sin(t))

\int \frac{sin ( t )}{1 + sin ( t )} d t

\frac{2}{tan ( \frac{t}{2} ) + 1} + t + C

Solution steps

\int \frac{sin ( t )}{1 + sin ( t )} d t

\frac{sin ( t )}{1 + sin ( t )} = - \frac{1}{1 + sin ( t )} + 1

= \int - \frac{1}{1 + sin ( t )} + 1 d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{1 + sin ( t )} d t + \int 1 d t

\int \frac{1}{1 + sin ( t )} d t = - \frac{2}{tan ( \frac{t}{2} ) + 1}

\int 1 d t = t

= - (- \frac{2}{tan ( \frac{t}{2} ) + 1}) + t

Simplify

= \frac{2}{tan ( \frac{t}{2} ) + 1} + t

Add a constant to the solution

= \frac{2}{tan ( \frac{t}{2} ) + 1} + t + C

What is the integral of (sin(t))/(1+sin(t)) ?
The integral of (sin(t))/(1+sin(t)) is 2/(tan(t/2)+1)+t+C