Aspire Faculty ID #18684 · Topic: NIMCET 2026 · Just now
NIMCET 2026

Evaluate the following definite integral:$\int_{0}^{\frac{\pi}{4}} \frac{dx}{\cos^4 x}$

Solution

$\int_{0}^{\frac{\pi}{4}} \frac{dx}{\cos^4 x}$ 
$=\int_{0}^{\frac{\pi}{4}} \sec^4 xdx$ 
Now, $\sec^4 x=\sec^2 x(1+\tan^2 x)$ 

Let $\tan x=t$ $\sec^2 xdx=dt$ 

Limits: 
$x=0 \Rightarrow t=0$ 
$x=\frac{\pi}{4} \Rightarrow t=1$ 

So, $\int_0^1 (1+t^2)dt$ 
$=\left[t+\frac{t^3}{3}\right]_0^1$ 
$=1+\frac{1}{3}$ 
$=\frac{4}{3}$
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