NIMCET 2008 — Mathematics PYQ

1. Understand the Property of Symmetric Limits

For any definite integral with symmetric limits $[-a,a]$, we use the following property:

A function $\varphi (x)$ is odd if $\varphi (-x)=-\varphi (x)$.

2. Define the Integrand

Let the entire function inside the integral be $\varphi (x)$:

3. Test for Symmetry (Even or Odd)

To check if $\varphi (x)$ is even or odd, we replace $x$ with $-x$:

Now, observe the second bracket $[g(-x)-g(x)]$. We can factor out a negative sign:

Substituting this back into our expression for $\varphi (-x)$:

4. Conclusion of the Integration

Since $\varphi (-x)=-\varphi (x)$, the function $\varphi (x)$ is an odd function.

According to the integral property for odd functions over symmetric limits:

Final Answer

The value of the integral is $0$.

Correct Option: (d)