What's that under the bridge?

Calculus Level 2

$\large{\int_{-\pi}^\pi \frac{x^4 (\sin^5 x - \sin^3 x - 2x)}{(x^2 + 5)(7 - \cos^4 x)}} \, dx = \, ?$

The answer is 0.

2 solutions

Prakhar Bindal
Jan 25, 2017

The function given is odd . hence its integral from -a to a would be zero

The first thing I wondered when I saw the function was that why was it in Level 2 with just mere 40 points to score. That exact moment I saw the limits and knew what such a clever integral was hiding within itself!

Tapas Mazumdar - 4 years, 4 months ago

I originally posted it in the Hard section because I wanted to make people sweat a little first :)

Steven Chase - 4 years, 3 months ago

If there is a polynomial $P$ , then you should prove that $\displaystyle \int ^ { a } _ { -a } P \quad dx = 0$ .

. . - 2 weeks, 6 days ago

. .
May 21, 2021

Let $P$ to a polynomial.

Then, $\displaystyle \int ^ { a } _ { - a } P \quad dx = \boxed { 0 }$ .

You dont have to expand those polynomials.

What's that under the bridge?

The answer is 0.

2 solutions

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