Question

Integral indefinida

$\int\ {} \frac{2~cotg~x-~3~sen^2x}{sen~x} \, dx$

Answer 1

$I=\displaystyle\int\!\frac{2\,\mathrm{cotg\,}x-3\,\mathrm{sen\,}x}{\mathrm{sen\,}x}\,dx\\\\\\ =\int\!\frac{2\cdot \frac{\cos x}{\mathrm{sen\,}x}-3\,\mathrm{sen\,}x}{\mathrm{sen\,}x}\,dx\\\\\\ =\int\!\left(\frac{2\cdot \frac{\cos x}{\mathrm{sen\,}x}}{\mathrm{sen\,}x}-\frac{3\,\mathrm{sen\,}x}{\mathrm{sen\,}x} \right )dx$


Na primeira fração para simplificar, vamos multiplicar o numerador e o denominador por $\mathrm{sen\,}x\ne 0:$

$=\displaystyle\int\!\left(\frac{2\cdot \frac{\cos x}{\mathrm{sen\,}x}}{\mathrm{sen\,}x}\cdot \frac{\mathrm{sen\,}x}{\mathrm{sen\,}x}-\frac{3\,\mathrm{sen\,}x}{\mathrm{sen\,}x} \right )dx\\\\\\ =\int\!\left(\frac{2\cdot \frac{\cos x}{\mathrm{sen\,}x}\cdot \mathrm{sen\,}x}{\mathrm{sen\,}x\cdot \mathrm{sen\,}x}-\frac{3\,\mathrm{sen\,}x}{\mathrm{sen\,}x} \right )dx\\\\\\ =\int\!\left(\frac{2\cdot \cos x}{\mathrm{sen^2\,}x}-\frac{3\,\mathrm{sen\,}x}{\mathrm{sen\,}x} \right )dx\\\\\\ =\int\!\left(2\cdot \frac{\cos x}{\mathrm{sen^2\,}x}-3 \right )dx$

$=\displaystyle 2\int\!\frac{1}{\mathrm{sen^2\,}x}\cdot \cos x\,dx-3\int\!dx\\\\\\ =\int\!\frac{1}{u^2}\,du-3x~~~~~~(u=\mathrm{sen\,}x)\\\\\\ =\int\!u^{-2}\,du-3x\\\\\\ =\frac{u^{-2+1}}{-2+1}-3x+C\\\\\\ =\frac{u^{-1}}{-1}-3x+C\\\\\\ =\,-\frac{1}{u}-3x+C\\\\\\ =-\,\frac{1}{\mathrm{sen\,}x}-3x+C$


$\therefore~~\boxed{\begin{array}{c}\displaystyle\int\!\frac{2\,\mathrm{cotg\,}x-3\,\mathrm{sen\,}x}{\mathrm{sen\,}x}\,dx=-\,\mathrm{cossec\,}x-3x+C \end{array}}$


Bons estudos! :-)

Answer 2

$\sf \displaystyle \int \frac{2cotg \left(x\right)-3 ~sen ^2\left(x\right)}{sen \left(x\right)}dx\\\\\\=\int \:-3sin \left(x\right)+\frac{2co\tan \left(x\right)}{\sin \left(x\right)}dx\\\\\\=-\int \:3\sin \left(x\right)dx+\int \frac{2co\tan \left(x\right)}{\sin \left(x\right)}dx\\\\\\=-\left(-3\cos \left(x\right)\right)+2co\ln \left|\tan \left(x\right)+\sec \left(x\right)\right|\\\\\\=3\cos \left(x\right)+2co\ln \left|\tan \left(x\right)+\sec \left(x\right)\right|$

$\sf \to \boxed{\sf =3\cos \left(x\right)+2co\ln \left|\tan \left(x\right)+\sec \left(x\right)\right|+C}$