The equation
`int_(-pi//4)^(pi//4) {a|sin x|+(b sin x)/(1+cos x)+c}dx=0`where a,b,c are constants, gives a relation between

Options

A. a,b and c
B. a and c
C. a and b
D. b and c

Answer

To solve the equation

\[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left( a |\sin x| + \frac{b \sin x}{1 + \cos x} + c \right) dx = 0, \]

we will break down the integral into manageable parts and evaluate each component step by step.

### Step 1: Break down the integral
The integral can be split into three parts:

\[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} a |\sin x| \, dx + \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{b \sin x}{1 + \cos x} \, dx + \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} c \, dx = 0. \]