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Evaluation of $\displaystyle \int\sqrt[4]{\tan x}dx$

$\bf{My\; Try}::$ Let $\tan x = t^4\;\;,$ Then $\sec^2 xdx = 4t^3dt$. So $\displaystyle dx = \frac{4t^3}{1+t^8}dt$

So Integral Convert into $\displaystyle 4\int\frac{t^4}{1+t^8}dt = 2\int \frac{(t^4+1)+(t^4-1)}{t^8+1}dt$

So Integral is $\displaystyle 2\int\frac{t^4+1}{t^8+1}dt+2\int\frac{t^4-1}{t^4+1}dt$

Now How can I solve after that

Help me

Thanks

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  • $\begingroup$ I get this very long and nasty expression from Wolf $\endgroup$ – Santosh Linkha Jul 8 '14 at 17:10
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    $\begingroup$ Using Wolf amounts to cheating :( . $\endgroup$ – Count Iblis Jul 8 '14 at 17:14
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    $\begingroup$ Here is the same problem. $\endgroup$ – Mhenni Benghorbal Jul 8 '14 at 18:21
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Let's compute $\int\frac{t^4}{t^8+1}dt$ using partial fractions.

We start with locating the poles of the integrand in the complex plane. They are at $t_n = \exp\left[\frac{(2 n+1) i\pi}{8}\right]$. Then near each pole the singular behavior of the integrand is:

$$t_n^4\left[\lim_{t\to t_n}\frac{t-t_n}{t^8 + 1}\right]\frac{1}{t-t_n}=\frac{1}{8 t_n^3}\frac{1}{t-t_n}$$

Then the sum of this over n from n = 0 to n = 7 is the partial fraction expansion (to see this, subtract this from the integrand, the resulting expression doesn't have any singularities anymore therefore it must be a polynomial, but it tends to zero at infinity so it is identical to zero).

To compute the integral, you can integrate this term by term. To convert the expression with complex terms to a manifestly real expression, you can take the terms involving $t_n$ and $t_{8-n}$ together, they are complex conjugates of each other, so you can replace them by twice the real part of the term involving $t_n$ (except for n = 4 in which case the term is real). So, we need to simplify:

$$\Re\frac{1}{4 t_n^3}\log(t-t_n)$$

This is easy, we have an expression of the form

$$\exp(i\alpha)\log(t-\exp(i\beta))$$

The real and imaginary parts of the logarithm are obtained as follows. You write the arguments as the sum of the real and imaginary parts:

$$\log\left[t - \cos(\beta) - i\sin(\beta)\right]$$

And using

$$x+i y= \sqrt{x^2 + y^2}\exp(i\arctan(\frac{y}{x}))$$

where I've assumed for simplicity that $\arctan(\frac{y}{x})$ yields the correct argument of the complex number.

So, obtaining the integral is straightforward using partial fractions.

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You are on the right track, but the remaining computations are tedious. First, note that we have the factorizations $$t^4 + 1 = (t^2 - \sqrt{2} t + 1)(t^2 + \sqrt{2} t + 1),$$ and $$t^8 + 1 = (t^2 - \alpha t + 1)(t^2 + \alpha t + 1)(t^2 - \beta t + 1)(t^2 + \beta t + 1),$$ where $\alpha = \sqrt{2+\sqrt{2}}$ and $\beta = \sqrt{2 - \sqrt{2}} = \sqrt{2}/\alpha$. These then admit partial fraction decomposition of the two integrands you obtained, from which we can then eventually arrive at an antiderivative.

Incidentally, the separation into two terms as you have done is not particularly useful.

An alternate approach would be to perform the partial fraction decomposition of $\dfrac{t^4}{t^8+1}$ into linear factors over $\mathbb C$, for which the antiderivative is easier to compute as these will be logarithms of the form $$\log (t - \zeta_{16}^{2k+1})$$ for $\zeta_{16} = e^{\pi i/8}$ a primitive $16^{\rm th}$ root of unity and $k = 0, 1, \ldots, 7$. Then we recombine these appropriately to obtain an expression without complex numbers.

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