Question

Integral de 1/x + 1/2x + cós(x) - 5^x dx

O resultado tem q dar 3/2 ln(x) + sen(x) - 5^x/ln(5) + c

Answer 1

∫ 1/x + 1/2x + cos(x) - 5^x dx = ∫ 1/x dx + ∫ 1/2x dx + ∫ cos(x) dx - ∫ 5^x dx
∫ 1/x dx = ln(x) + c 
∫ 1/2x dx = 1/2 ∫ 1/x dx = 1/2 ln(x) + c
∫ cos(x) dx = sen(x) + c
∫ 5^x dx = 5^x/ln(5) + c pois ∫ a^x dx = a^x/ln(a) + c

Logo, temos:
ln(x)+1/2 ln(x) + sen(x) - 5^x/ln(5) + c
3/2 ln(x) + sen(x) - 5^x/ln(5) + c

Answer 2

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$\displaystyle\sf{\int\left(\dfrac{1}{x}+\dfrac{1}{2x}+cos(x)-5^x\right)dx=\ell n|x|+\dfrac{1}{2}\ell n|x|+sen(x)-\dfrac{5^x}{\ell n(5)}+k}\\\displaystyle\sf{\int\left(\dfrac{1}{x}+\dfrac{1}{2x}+cos(x)-5^x\right)dx=\dfrac{3}{2}\ell n|x|+sen(x)-\dfrac{5^x}{\ell n(5)}+k}$