Question

Evaluate \[ \int_{0}^{\frac{\pi}{4}}\frac{dx}{\cos^4x}. \]

• A. \(1\)
• B. \(\frac43\)
• C. \(\frac13\)
• D. \(\frac23\)

Hint:

For integrals involving \(\sec^4x\), the substitution \(t=\tan x\) immediately converts the integral into a polynomial integral.

Answer 1

The Correct Option is B

Concept: Since \[ \frac1{\cos^4x} = \sec^4x, \] we use substitution \[ t=\tan x. \] Then \[ dt=\sec^2x\,dx. \]

Step 1: Transform the integral.
\[ I=\int_0^{\pi/4}\sec^4x\,dx \] \[ =\int_0^{\pi/4}\sec^2x\cdot\sec^2x\,dx \] Let \[ t=\tan x. \] Then \[ dt=\sec^2x\,dx. \] Also, \[ \sec^2x=1+\tan^2x=1+t^2. \] Therefore \[ I=\int_0^1(1+t^2)\,dt. \]

Step 2: Integrate.
\[ I= \left[t+\frac{t^3}{3}\right]_0^1 \] \[ = 1+\frac13 \] \[ = \frac43. \] Hence \[ \boxed{\frac43} \]