NDA 2026 — Mathematics PYQ

To solve this integral, we first simplify the expression using the given relations and then apply the property of definite integrals: ${\int }_0^{a}h(x)dx={\int }_0^{a}h(a-x)dx$.

Step 1: Simplify $g(4-x)$

Given $g(x)=f(x)+f(4-x)$, we substitute $4-x$ for $x$:

$$g(4-x)=f(4-x)+f(4-(4-x))$$

$$g(4-x)=f(4-x)+f(x)$$

Step 2: Rewrite the integral

The integral $I={\int }_0^4\frac{f(4-x)}{g(4-x)}dx$ becomes:

$$I={\int }_0^4\frac{f(4-x)}{f(4-x)+f(x)}dx$$

Step 3: Apply the definite integral property

Using the property ${\int }_0^{4}h(x)dx={\int }_0^{4}h(4-x)dx$:

$$I={\int }_0^4\frac{f(4-(4-x))}{f(4-(4-x))+f(4-x)}dx$$

$$I={\int }_0^4\frac{f(x)}{f(x)+f(4-x)}dx$$

We know that ${\int }_0^4\frac{f(x)}{f(x)+f(4-x)}dx=2$. Therefore:

$$I=2$$

The value of the integral is $2$, and the correct option is (c).