[tex]\int\ (\frac{e^x}{2}+x\sqrt{x}).dx \\\\\int\ \frac{e^x}{2}.dx+\int x\sqrt{x}.dx\\\\\frac{1}{2}\int e^x.dx+\int x.x^{\frac{1}{2}}.dx\\\\\frac{1}{2}\int e^x.dx+\int x^{\frac{3}{2}}.dx\\\\\frac{e^x}{2}+C_1+\frac{2}{5}x^{\frac{5}{2}}+C_2\\\\\frac{e^x}{2}+\frac{2x^{\frac{5}{2}}}{5}+C[/tex]
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[tex]\int(2e^u+\frac{6}{u}+\ln2).du\\\\\int 2e^u.du+\int \frac{6}{u}.du+\int \ln2.du\\\\2\int e^u.du+6\int u^{-1}+\ln2\int du\\\\2e^u+C_1+6\ln|u|+C_2+\ln2.u+C_3\\\\2e^u+6\ln|u|+\ln2.u+C[/tex]
Prova diferenciando as funções:
[tex]\frac{d}{dx}(\frac{e^x}{2}+\frac{2x^{\frac{5}{2}}}{5}+C)\\\\\frac{d}{dx}(\frac{e^x}{2})+\frac{d}{dx}(\frac{2x^{\frac{5}{2}}}{5})+\frac{d}{dx}(C)\\\\\frac{e^x}{2}+\frac{5}{2}.\frac{2}{5}.x^{\frac{3}{2}}+0\\\\\frac{e^x}{2}+x^{\frac{3}{2}}\\\\\frac{e^x}{2}+{x\sqrt{x}}[/tex]
[tex]\frac{d}{du}(2e^u+6\ln|u|+\ln2.u+C)\\\\\frac{d}{du}(2e^u)+\frac{d}{du}(6.ln|u|)+\frac{d}{du}(\ln2.u)+\frac{d}{du}(c)\\\\2e^u+\frac{6}{u}+\ln2+0[/tex]
