Question

To determine the value of the integral function.

Short answer

The value of the integral is \(0\).

Step-by-step solution

Given data

The integral function is \(\int_{ - \pi /4}^{\pi /4} {\frac{{{t^4}\tan t}}{{2 + \cos t}}} dt\)

The region lies between \(x = - \frac{\pi }{4}\) and \(x = \frac{\pi }{4}\).

Concept used of Integral and the Theorem used

Integral of a function \(f(x)\) is denoted as \(\int_a^b f (x)dx\).

Theorem:

a) If\(f( - t) = f(t)\), that is the equation of the curve is unchanged, when\(t\)is replaced by\(( - t)\), then the function is an even function and the curve is symmetric about\(y\)-axis.

b) If \(f( - t) = f(t)\), that is the equation of the curve is changed, when \(t\) is replaced by \(( - t)\), then the function is an odd function and the curve is symmetric about the origin.

Simplify the function

The expression of the function \(f(t)\) is shown below:

\(f(t) = \frac{{{t^4}\tan t}}{{2 + \cos t}}\)

Substitute \( - t\) for \(t\) to check whether the function is odd or even as shown below:

\(\begin{array}{l}f( - t) = \frac{{{{( - t)}^4}\tan ( - t)}}{{2 + \cos ( - t)}}\\ = \frac{{{t^4}( - \tan t)}}{{2 + \cos t}}\\ = - \frac{{{t^4}\tan t}}{{2 + \cos t}}\\ = - f(t)\end{array}\)

Apply the Theorem

Apply the Theorem:

a) If \(f( - t) = f(t)\), that is the equation of the curve is unchanged, when \(t\) is replaced by \( - t\), then the function is an even function and the curve is symmetric about \(y\)-axis.

b) If \(f( - t) = - f(t)\), that is the equation of the curve is changed, when \(t\) is replaced by \( - t\), then the function is an odd function and the curve is symmetric about the origin.

Refer to Theorem.

In accordance to Theorem, the given function is an odd function, so the integral value of the function is 0.

Therefore, the value of the integral is. 0 .