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a) 
![=\left[\dfrac{x^{4}}{4}-x^{2} \right ]_{-1}^{2}\\ \\ \\ =\left[\dfrac{2^{4}}{4}-2^{2} \right ]-\left[\dfrac{(-1)^{4}}{4}-(-1)^{2} \right ]\\ \\ \\ =\left[\dfrac{16}{4}-4 \right ]-\left[\dfrac{1}{4}-1 \right ]\\ \\ \\ =0-\left[\dfrac{1-4}{4} \right ]\\ \\ \\ =0-\left[-\dfrac{3}{4} \right ]\\ \\ \\ =\dfrac{3}{4} =\left[\dfrac{x^{4}}{4}-x^{2} \right ]_{-1}^{2}\\ \\ \\ =\left[\dfrac{2^{4}}{4}-2^{2} \right ]-\left[\dfrac{(-1)^{4}}{4}-(-1)^{2} \right ]\\ \\ \\ =\left[\dfrac{16}{4}-4 \right ]-\left[\dfrac{1}{4}-1 \right ]\\ \\ \\ =0-\left[\dfrac{1-4}{4} \right ]\\ \\ \\ =0-\left[-\dfrac{3}{4} \right ]\\ \\ \\ =\dfrac{3}{4}](https://tex.z-dn.net/?f=%3D%5Cleft%5B%5Cdfrac%7Bx%5E%7B4%7D%7D%7B4%7D-x%5E%7B2%7D+%5Cright+%5D_%7B-1%7D%5E%7B2%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cdfrac%7B2%5E%7B4%7D%7D%7B4%7D-2%5E%7B2%7D+%5Cright+%5D-%5Cleft%5B%5Cdfrac%7B%28-1%29%5E%7B4%7D%7D%7B4%7D-%28-1%29%5E%7B2%7D+%5Cright+%5D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cdfrac%7B16%7D%7B4%7D-4+%5Cright+%5D-%5Cleft%5B%5Cdfrac%7B1%7D%7B4%7D-1+%5Cright+%5D%5C%5C+%5C%5C+%5C%5C+%3D0-%5Cleft%5B%5Cdfrac%7B1-4%7D%7B4%7D+%5Cright+%5D%5C%5C+%5C%5C+%5C%5C+%3D0-%5Cleft%5B-%5Cdfrac%7B3%7D%7B4%7D+%5Cright+%5D%5C%5C+%5C%5C+%5C%5C+%3D%5Cdfrac%7B3%7D%7B4%7D)
b)
![=\left[u+\dfrac{1}{2}\cdot \dfrac{u^{5}}{5}-\dfrac{2}{5}\cdot \dfrac{u^{10}}{10} \right ]_{0}^{1}\\ \\ \\ =\left[u+\dfrac{u^{5}}{10}-\dfrac{u^{10}}{25} \right ]_{0}^{1}\\ \\ \\ =\left[1+\dfrac{1^{5}}{10}-\dfrac{1^{10}}{25} \right ]-\left[0+\dfrac{0^{5}}{10}-\dfrac{0^{10}}{25} \right ]\\ \\ \\ =\left[1+\dfrac{1}{10}-\dfrac{1}{25} \right ]-0\\ \\ \\ =\dfrac{50+5-2}{50}\\ \\ \\ =\dfrac{53}{50} =\left[u+\dfrac{1}{2}\cdot \dfrac{u^{5}}{5}-\dfrac{2}{5}\cdot \dfrac{u^{10}}{10} \right ]_{0}^{1}\\ \\ \\ =\left[u+\dfrac{u^{5}}{10}-\dfrac{u^{10}}{25} \right ]_{0}^{1}\\ \\ \\ =\left[1+\dfrac{1^{5}}{10}-\dfrac{1^{10}}{25} \right ]-\left[0+\dfrac{0^{5}}{10}-\dfrac{0^{10}}{25} \right ]\\ \\ \\ =\left[1+\dfrac{1}{10}-\dfrac{1}{25} \right ]-0\\ \\ \\ =\dfrac{50+5-2}{50}\\ \\ \\ =\dfrac{53}{50}](https://tex.z-dn.net/?f=%3D%5Cleft%5Bu%2B%5Cdfrac%7B1%7D%7B2%7D%5Ccdot+%5Cdfrac%7Bu%5E%7B5%7D%7D%7B5%7D-%5Cdfrac%7B2%7D%7B5%7D%5Ccdot+%5Cdfrac%7Bu%5E%7B10%7D%7D%7B10%7D+%5Cright+%5D_%7B0%7D%5E%7B1%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5Bu%2B%5Cdfrac%7Bu%5E%7B5%7D%7D%7B10%7D-%5Cdfrac%7Bu%5E%7B10%7D%7D%7B25%7D+%5Cright+%5D_%7B0%7D%5E%7B1%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B1%2B%5Cdfrac%7B1%5E%7B5%7D%7D%7B10%7D-%5Cdfrac%7B1%5E%7B10%7D%7D%7B25%7D+%5Cright+%5D-%5Cleft%5B0%2B%5Cdfrac%7B0%5E%7B5%7D%7D%7B10%7D-%5Cdfrac%7B0%5E%7B10%7D%7D%7B25%7D+%5Cright+%5D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B1%2B%5Cdfrac%7B1%7D%7B10%7D-%5Cdfrac%7B1%7D%7B25%7D+%5Cright+%5D-0%5C%5C+%5C%5C+%5C%5C+%3D%5Cdfrac%7B50%2B5-2%7D%7B50%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cdfrac%7B53%7D%7B50%7D)
c)
![=\displaystyle\int\limits_{1}^{8}{x^{1/3}\,dx}\\ \\ \\ =\left[\dfrac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1} \right ]_{1}^{8}\\ \\ \\ =\left[\dfrac{x^{4/3}}{\frac{4}{3}} \right ]_{1}^{8}\\ \\ \\ =\left[\dfrac{3x^{4/3}}{4} \right ]_{1}^{8}\\ \\ \\ =\left[\dfrac{3(\,^{3}\!\!\!\sqrt{x})^{4}}{4} \right ]_{1}^{8}\\ \\ \\ =\left[\dfrac{3(\,^{3}\!\!\!\sqrt{8})^{4}}{4} \right ]-\left[\dfrac{3(\,^{3}\!\!\!\sqrt{1})^{4}}{4} \right ]\\ \\ \\ =\left[\dfrac{3\cdot 2^{4}}{4} \right ]-\left[\dfrac{3\cdot 1^{4}}{4} \right ]\\ \\ \\ =12-\dfrac{3}{4}\\ \\ \\ =\dfrac{45}{4} =\displaystyle\int\limits_{1}^{8}{x^{1/3}\,dx}\\ \\ \\ =\left[\dfrac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1} \right ]_{1}^{8}\\ \\ \\ =\left[\dfrac{x^{4/3}}{\frac{4}{3}} \right ]_{1}^{8}\\ \\ \\ =\left[\dfrac{3x^{4/3}}{4} \right ]_{1}^{8}\\ \\ \\ =\left[\dfrac{3(\,^{3}\!\!\!\sqrt{x})^{4}}{4} \right ]_{1}^{8}\\ \\ \\ =\left[\dfrac{3(\,^{3}\!\!\!\sqrt{8})^{4}}{4} \right ]-\left[\dfrac{3(\,^{3}\!\!\!\sqrt{1})^{4}}{4} \right ]\\ \\ \\ =\left[\dfrac{3\cdot 2^{4}}{4} \right ]-\left[\dfrac{3\cdot 1^{4}}{4} \right ]\\ \\ \\ =12-\dfrac{3}{4}\\ \\ \\ =\dfrac{45}{4}](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint%5Climits_%7B1%7D%5E%7B8%7D%7Bx%5E%7B1%2F3%7D%5C%2Cdx%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cdfrac%7Bx%5E%7B%5Cfrac%7B1%7D%7B3%7D%2B1%7D%7D%7B%5Cfrac%7B1%7D%7B3%7D%2B1%7D+%5Cright+%5D_%7B1%7D%5E%7B8%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cdfrac%7Bx%5E%7B4%2F3%7D%7D%7B%5Cfrac%7B4%7D%7B3%7D%7D+%5Cright+%5D_%7B1%7D%5E%7B8%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cdfrac%7B3x%5E%7B4%2F3%7D%7D%7B4%7D+%5Cright+%5D_%7B1%7D%5E%7B8%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cdfrac%7B3%28%5C%2C%5E%7B3%7D%5C%21%5C%21%5C%21%5Csqrt%7Bx%7D%29%5E%7B4%7D%7D%7B4%7D+%5Cright+%5D_%7B1%7D%5E%7B8%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cdfrac%7B3%28%5C%2C%5E%7B3%7D%5C%21%5C%21%5C%21%5Csqrt%7B8%7D%29%5E%7B4%7D%7D%7B4%7D+%5Cright+%5D-%5Cleft%5B%5Cdfrac%7B3%28%5C%2C%5E%7B3%7D%5C%21%5C%21%5C%21%5Csqrt%7B1%7D%29%5E%7B4%7D%7D%7B4%7D+%5Cright+%5D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cdfrac%7B3%5Ccdot+2%5E%7B4%7D%7D%7B4%7D+%5Cright+%5D-%5Cleft%5B%5Cdfrac%7B3%5Ccdot+1%5E%7B4%7D%7D%7B4%7D+%5Cright+%5D%5C%5C+%5C%5C+%5C%5C+%3D12-%5Cdfrac%7B3%7D%7B4%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cdfrac%7B45%7D%7B4%7D)
d)
![=\displaystyle\int\limits_{1}^{2}{\left(\dfrac{v^{3}}{v^{4}}+\dfrac{3v^{6}}{v^{4}} \right )\,dv}\\ \\ \\ =\displaystyle\int\limits_{1}^{2}{\left(\dfrac{1}{v}+3v^{2} \right )\,dv}\\ \\ \\ =\left[\mathrm{\ell n\,}|v|+v^{3}\right]_{1}^{2}\\ \\ =\left[\mathrm{\ell n\,}|2|+2^{3}\right]-\left[\mathrm{\ell n\,}|1|+1^{3}\right]\\ \\ =\left[\mathrm{\ell n\,}(2)+8\right]-\left[0+1\right]\\ \\ =\mathrm{\ell n\,}(2)+8-1\\ \\ =\mathrm{\ell n\,}(2)+7 =\displaystyle\int\limits_{1}^{2}{\left(\dfrac{v^{3}}{v^{4}}+\dfrac{3v^{6}}{v^{4}} \right )\,dv}\\ \\ \\ =\displaystyle\int\limits_{1}^{2}{\left(\dfrac{1}{v}+3v^{2} \right )\,dv}\\ \\ \\ =\left[\mathrm{\ell n\,}|v|+v^{3}\right]_{1}^{2}\\ \\ =\left[\mathrm{\ell n\,}|2|+2^{3}\right]-\left[\mathrm{\ell n\,}|1|+1^{3}\right]\\ \\ =\left[\mathrm{\ell n\,}(2)+8\right]-\left[0+1\right]\\ \\ =\mathrm{\ell n\,}(2)+8-1\\ \\ =\mathrm{\ell n\,}(2)+7](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cint%5Climits_%7B1%7D%5E%7B2%7D%7B%5Cleft%28%5Cdfrac%7Bv%5E%7B3%7D%7D%7Bv%5E%7B4%7D%7D%2B%5Cdfrac%7B3v%5E%7B6%7D%7D%7Bv%5E%7B4%7D%7D+%5Cright+%29%5C%2Cdv%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cdisplaystyle%5Cint%5Climits_%7B1%7D%5E%7B2%7D%7B%5Cleft%28%5Cdfrac%7B1%7D%7Bv%7D%2B3v%5E%7B2%7D+%5Cright+%29%5C%2Cdv%7D%5C%5C+%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cmathrm%7B%5Cell+n%5C%2C%7D%7Cv%7C%2Bv%5E%7B3%7D%5Cright%5D_%7B1%7D%5E%7B2%7D%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cmathrm%7B%5Cell+n%5C%2C%7D%7C2%7C%2B2%5E%7B3%7D%5Cright%5D-%5Cleft%5B%5Cmathrm%7B%5Cell+n%5C%2C%7D%7C1%7C%2B1%5E%7B3%7D%5Cright%5D%5C%5C+%5C%5C+%3D%5Cleft%5B%5Cmathrm%7B%5Cell+n%5C%2C%7D%282%29%2B8%5Cright%5D-%5Cleft%5B0%2B1%5Cright%5D%5C%5C+%5C%5C+%3D%5Cmathrm%7B%5Cell+n%5C%2C%7D%282%29%2B8-1%5C%5C+%5C%5C+%3D%5Cmathrm%7B%5Cell+n%5C%2C%7D%282%29%2B7)
b)
c)
d)
matematicando:
Pq vc resolveu ln(1)=0 e ln(2) n resolvveu ?e pq ln(1) é igual a zero ?
mas ln(2) é um número real
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