The GATE Grind

GATE 2025 IN – Question 38

Engineering Mathematics · Calculus and vector calculus · 2 marks · Multiple choice

The value of $\displaystyle\int_{-\pi}^{\pi}(\cos^6x+\cos^4x)\,dx$ is

  1. $\pi/2$
  2. $5\pi/8$
  3. $11\pi/8$
  4. $9\pi/8$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) $11\pi/8$

Explanation

Over a complete period, the means of even cosine powers are $\langle\cos^{2m}x\rangle=\binom{2m}{m}/2^{2m}$.

Thus $\langle\cos^6x\rangle=20/64=5/16$ and $\langle\cos^4x\rangle=6/16=3/8$.

The integral is $2\pi(5/16+3/8)=11\pi/8$, **C**.