I am trying to evaluate the definite integral of (ex. 36 pg. 523, James Stewart Calculus 7e):
$$\int_{-\frac {\pi}{4}}^{\frac {\pi}{4}} \frac {x^2 \tan x}{1+{\cos^4{x}}}dx$$
However, I have not got any possible approach recently. I have tried using trigonometric substitution but $x^2$ keep challenging me. I would be grateful if there is any suggestions, thank you!
Evaluate $\int_{-\frac {\pi}{4}}^{\frac {\pi}{4}} \frac {x^2 \tan x}{1+{\cos^4{x}}}dx$
calculusdefinite integrals
Best Answer
Take the function, I will call it $f$, and plug in $-x$ for $x$. What do you get? You get exactly $-f(x)$. Now notice that you are integrating over the interval $(-\frac{\pi}{4},\frac{\pi}{4})$. That which you will add in the positive region, you will substract by integrating in the negative region. Hence the result is $0$.