Integral tan^4 x dx


Menentukan \int {{{\tan }^4}x} {\mathop{\rm dx}\nolimits}

Turunan dari \frac{1}{3}{\tan ^3}x = {\tan ^2}x.\frac{{d(\tan x)}}{{dx}} = {\tan ^2}x.{\sec ^2}x

= {\tan ^2}x({\tan ^2}x + 1) = {\tan ^4}x + {\tan ^2}x

Jadi integral dari {\tan ^4}x + {\tan ^2}x adalah

\int {({{\tan }^4}x + {{\tan }^2}x)dx = \frac{1}{3}{{\tan }^3}}  + C

maka
\int {{{\tan }^4}x} {\mathop{\rm dx}\nolimits}
= \int {({{\tan }^4}x + {{\tan }^2}x - {{\tan }^2}x){\mathop{\rm dx}\nolimits} }
= \int {({{\tan }^4}x + {{\tan }^2}x){\mathop{\rm dx}\nolimits} }  - \int {{{\tan }^2}x} {\mathop{\rm dx}\nolimits}
= \frac{1}{3}{\tan ^3}x - \int {({{\sec }^2}x - 1)} {\mathop{\rm dx}\nolimits}
= \frac{1}{3}{\tan ^3}x - \tan x + x + C

Semoga bermanfaat.

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