Find: \int \frac{\tan ^{2}x\, \sec ^{2}x}{1-\tan ^{6}x}\, dx.
 

 

 

 

 
 
 
 
 

Answers (1)

I= \int \frac{\tan ^{2}x\, \sec ^{2}x}{1-\tan ^{6}x}\,\, dx
\Rightarrow \, \, I= \int \frac{\tan ^{2}x\, \sec ^{2}x}{1-(\tan ^{3}x)^{2}}\,\, dx

put \tan ^{3}x= t
differntiating 3\tan ^{2}x\, \sec ^{2}dx= dt 
w.r.t x
\tan ^{2}x\, \sec ^{x}dx=\frac{dt}{3}
\Rightarrow I=\frac{1}{3}\int \frac{dt}{1-t^{2}}
= \frac{1}{3}\times \frac{1}{2\times 1} \log \left | \frac{1+t}{1-t} \right |+c
= \frac{1}{6}\, \log \left | \frac{1+\tan ^{3}x}{1-\tan ^{3}x} \right |+c

 

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