Question

Integral indefinida

$\int\ {}(5~cos x - 4~sen x) \, dx$

Answer 1

Lembremos de duas primitivas imediatas:

$\displaystyle\bullet\;\;\int\!\cos x=\mathrm{sen\,}x+C\\\\\\ \bullet\;\;\int\!\mathrm{sen\,} x=-\cos x+C$

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Então,

$\displaystyle\int\!(5\cos x-4\mathrm{sen\,}x)\,dx\\\\\\ =\int\!5\cos x\,dx+\int\!(-4)\,\mathrm{sen\,}x\,dx\\\\\\ =5\int\!\cos x\,dx-4\int\!\mathrm{sen\,}x\,dx\\\\\\ =5\cdot \mathrm{sen\,}x-4\cdot (-\cos x)+C\\\\\\ =5\,\mathrm{sen\,}x+4\cos x+C\\\\\\\\ \therefore~~\boxed{\begin{array}{c}\displaystyle\int\!(5\cos x-4\,\mathrm{sen\,}x)\,dx=5\,\mathrm{sen\,}x+4\cos x+C \end{array}}$


Bons estudos! :-)

Answer 2

$\sf \displaystyle \int \:5cos \left(x\right)-4sin \left(x\right)dx\\\\\\{Aplique\:a\:regra\:da\:soma}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\\\\\=5\sin \left(x\right)-\left(-4~cos \left(x\right)\right)\\\\\\=5\sin \left(x\right)+4~cos \left(x\right)\\\\\\\to \boxed{\sf =5sin \left(x\right)+4cos \left(x\right)+C}$