Question

integration of dx/(12+12 cos x)​

Answer 1

$\rm{\displaystyle\int \frac{dx}{12+12\cos \left(x\right)}}$

• Factor $\sf{1/(12+12\cos \left(x\right))}$

$\rm{=\displaystyle\int \frac{1}{12+12\cos \left(x\right)}dx}$

• Take the Constant Out : $\sf{\int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx}$

$\rm{\displaystyle =\frac{1}{12}\cdot \int \frac{1}{\cos \left(x\right)+1}dx}$

• Apply u - Substitution : $\sf{u=\tan \left(x/2\right)}$

$\rm{\displaystyle=\frac{1}{12}\cdot \int \:1du}$

• Integral of a constant $\sf{ \int adx=ax}$

$\rm{\displaystyle =\frac{1}{12}\cdot \:1\cdot \:u}$

• Substitute back : $\sf{u=\tan \left(x/2\right)}$

$\rm{\displaystyle =\frac{1}{12}\cdot \:1\cdot \tan \left(\frac{x}{2}\right)}$

$\rm{\displaystyle=\frac{1}{12}\tan \left(\frac{x}{2}\right)}$

• Add a Constant to the Solution

$\rm{\displaystyle =\frac{1}{12}\tan \left(\frac{x}{2}\right)+C}$