Question

Integral indefinida

$\int\ {}(2e^2 -\frac{sen x}{cos x}+ \frac{2}{x^7}) \, dx$

Answer 1

$\displaystyle\int\!\left(2e^2-\frac{\mathrm{sen\,}x}{\cos x}+\frac{2}{x^7} \right )dx\\\\\\ =\int 2e^2\,dx+\int\frac{1}{\cos x}\cdot (-\mathrm{sen\,}x)\,dx+\int 2x^{-7}\,dx\\\\\\ =2\int e^2\,dx+\int\frac{1}{\cos x}\cdot (-\mathrm{sen\,}x)\,dx+2\int x^{-7}\,dx\\\\\\ =2e^2\,x+\int\frac{1}{u}\,du+2\cdot \frac{x^{-7+1}}{-7+1}~~~~~~(\text{Substitui\c{c}\~ao }u=\cos x)\\\\\\ =2e^2\,x+\mathrm{\ell n}|u|+2\cdot \frac{x^{-6}}{-6}+C$

$=2e^2\,x+\mathrm{\ell n}\left|\cos x\right|-\dfrac{1}{3}\,x^{-6}+C\\\\\\\\ \therefore~~\boxed{\begin{array}{c} \displaystyle\int\!\left(2e^2-\frac{\mathrm{sen\,}x}{\cos x}+\frac{2}{x^7} \right )dx=2e^2\,x+\mathrm{\ell n}\left|\cos x\right|-\dfrac{1}{3x^6}+C \end{array}}$


Bons estudos! :-)