Evaluate -tan 2xtan 5xtan 7x dx

It is known that, tan( 7x) = tan( 2x + 5x) tan 7x = tan 2x + tan 5x1 tan 2xtan 5x tan 7x tan 7xtan 2xtan 5x = tan 2x + tan 5x tan 2xtan 5 ...

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Standard XII

Mathematics

Standard Formulae - 1

Evaluate ∫t...

Question

Evaluate $\int tan 2 x tan 5 x tan 7 x d x$

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Solution

It is known that,
$tan (7 x) = tan (2 x + 5 x)$
$tan 7 x = \frac{tan 2 x + tan 5 x}{1 - tan 2 x tan 5 x}$
$tan 7 x - tan 7 x tan 2 x tan 5 x = tan 2 x + tan 5 x$
$tan 2 x tan 5 x tan 7 x = tan 7 x - tan 2 x - tan 5 x$
Integrating both sides,
$\int (tan 2 x tan 5 x tan 7 x) d x = \int (tan 7 x - tan 2 x - tan 5 x) d x$
$\int (tan 2 x tan 5 x tan 7 x) d x = \frac{1}{7} log | sec 7 x | - \frac{1}{2} log | sec 2 x | - \frac{1}{5} log | sec 5 x | + c$

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Evaluate ∫tan2xtan5xtan7xdx

It is known that,tan(7x)=tan(2x+5x)tan7x=tan2x+tan5x1−tan2xtan5xtan7x−tan7xtan2xtan5x=tan2x+tan5xtan2xtan5xtan7x=tan7x−tan2x−tan5xIntegrating both sides,∫(tan2xtan5xtan7x)dx=∫(tan7x−tan2x−tan5x)dx∫(tan2xtan5xtan7x)dx=17log|sec7x|−12log|sec2x|−15log|sec5x|+c

Evaluate $\int tan 2 x tan 5 x tan 7 x d x$