How do you integrate $\frac{tan (x)}{x}$ $(tan(x))/x$ ?

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Answer 1 · 2016-11-04T21:49:39

I don't believe there is an intrinsic function that is the anti-derivative

The power series solution is:
$int tanx/xdx = x+1/9x^3+2/75x^5-17/2205x^7+62/25515x^9+...$

I believe the only way to handle this integral is to use the Maclaurin power series for $tanx$ ; as follows;

$int tanx/xdx = int (x+1/3x^3+2/15x^5-17/315x^7+62/2835x^9+... )/xdx$

$:. int tanx/xdx = int 1+1/3x^2+2/15x^4-17/315x^6+62/2835x^8+...$

$:. int tanx/xdx = x+1/3x^3/3+2/15x^5/5-17/315x^7/7+62/2835x^9/9+...$

$:. int tanx/xdx = x+1/9x^3+2/75x^5-17/2205x^7+62/25515x^9+...$

How do you integrate (tan(x))/xtan(x)x(tan(x))/x?