NIMCET 2026 — Mathematics PYQ

To solve this integral, we first simplify the integrand using trigonometric identities.

1. Simplify the Integrand:

We know that $\frac{1}{{\cos }^{4}x}={\sec }^{4}x$. We can rewrite ${\sec }^{4}x$ as:

$${\sec }^{4}x={\sec }^{2}x\cdot {\sec }^{2}x=(1+{\tan }^{2}x){\sec }^{2}x$$

2. Set up the Integral:

The integral becomes:

$${\int }_0^{\frac{\pi }{4}}(1+{\tan }^{2}x){\sec }^{2}xdx$$

Let $u=\tan x$. Then, the differential $du={\sec }^{2}xdx$.

For the limits:

• When $x=0$, $u=\tan (0)=0$.

• When $x=\frac{\pi }{4}$, $u=\tan (\frac{\pi }{4})=1$.

3. Evaluate the Integral:

Substitute $u$ and $du$ into the integral:

$${\int }_0^1(1+u^2)du$$

Now, integrate with respect to $u$:

$${\left[u+\frac{u^3}{3}\right]}_0^1$$

Substitute the limits:

$$\left(1+\frac{1^3}{3}\right)-(0+0)=1+\frac{1}{3}=\frac{4}{3}$$

Correct Option: (b)