let >tanx=t then >sec^2x dx=dt or(1+tan^2x)dx=dt ∴∫ (tanx(1+tan^2x)/1+tanx)dx =∫(t/1+t)dt =∫(1+t 1/1+t)dt =∫ 1 dt ∫(t/1+t)dt ...

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Integration of Trigonometric Functions

∫tanx+tan3x/1...

Question

$\int \frac{t a n x + t a n^{3} x}{1 + t a n x} d x$

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Solution

$l e t t a n x = t t h e n s e c^{2} x d x = d t o r (1 + t a n^{2} x) d x = d t ∴ \int (\frac{t a n x (1 + t a n^{2} x)}{1 + t a n x}) d x = \int (\frac{t}{1 + t}) d t = \int (\frac{1 + t - 1}{1 + t}) d t = \int 1 d t - \int (\frac{t}{1 + t}) d t = t - l o g | 1 + t | + c = t a n x - l o g | 1 + t a n x | + c$

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