Question

alguém por favor
$\int\limi tan^{5} x dx$
sabendo que 1+ tan²=sec²x

Answer 1

          $\displaystyle I=\int\tan^5x\;dx\\ \\ \\ u=\tan x\to du = \sec^2 x \;dx\\ du = (1+\tan^2x)\;dx\\ \\ dx =\dfrac{du}{1+u^2}\\ \\ \\ I=\int u^5\cdot \dfrac{du}{1+u^2}\\ \\ \\ I=\int u^3-u +\dfrac{u}{1+u^2}\;du\\ \\ \\ I=\dfrac{u^4}{4}-\dfrac{u^2}{2}+\dfrac{1}{2}\int \dfrac{d(1+u^2)}{1+u^2}\\ \\ \\ I=\dfrac{u^4}{4}-\dfrac{u^2}{2}+\dfrac{1}{2}\ln |1+u^2|+C\\ \\ \\ \boxed{I=\dfrac{\tan^4x}{4}-\dfrac{\tan^2x}{2}+\dfrac{1}{2}\ln |1+\tan^2x|+C}$