Question

Encuentre todas las funciones f tales que
f'(x)=8sen(x)+(2x^5-√x)/x


Espero me colaboren, Gracias!!

Answer 1

$\displaystyle f^{\prime }(x)=8\sin x +\dfrac{2x^5-\sqrt{x}}{x} \\[6pt] f^{\prime }(x)=8\sin x +\dfrac{2x^5}{x}-\dfrac{x^{1/2}}{x} \\[6pt] f^{\prime }(x)=8\sin x +2x^4-x^{-1/2} \\[6pt] f(x)=\int (8\sin x +2x^4-x^{-1/2})\,dx\\[6pt] f(x)=8\int \sin x \,dx +2\int x^4\,dx-\int x^{-1/2}\,dx \ ..... \ (i)\\[6pt] \text{F\'ormula : } \ \int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C \quad , \quad n\neq -1 \\[6pt] \text{Adem\'as por tabla } \ \int \sin x\,dx =-\cos x+C$

$\text{En }(i) \\[6pt] f(x)=8(-\cos x)+2\cdot \dfrac{x^{4+1}}{4+1}-\dfrac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} +C \\[6pt] \boxed{f(x)=-8\cos x+\dfrac{2x^5}{5}-2x^{1/2}+C}$