Calcolare i seguenti integrali:

a) $\displaystyle  \int (x^3+5x+2) dx=\int x^3 dx + 5\int x^2 dx + \int 2 dx =$

$\displaystyle  =\frac{1}{4}x^{4}+\frac{5}{2}x^2+2x+c$

b) $\displaystyle  \int (\sqrt{x}+\sin x +e^x) dx= \int \sqrt{x} dx + \int \sin x dx + \int e^x dx=$

$\displaystyle  =\frac{1}{\frac{3}{2}}x^{\frac{3}{2}}-\cos x +e^x+c =\frac{2}{3}x^{\frac{3}{2}}-\cos x +e^x+c=$

$\displaystyle  = \frac{2}{3}+\sqrt{x^3}-\cos x +e^x+c $

c) $\displaystyle \int (\sqrt[3]{x}+\cos x) dx=\int \sqrt[3]{x} dx+\int \cos x dx=$

$\displaystyle =\frac{1}{\frac{4}{3}}x^{\frac{4}{3}}+\sin x + c=\frac{3}{4}x^{\frac{4}{3}}+\sin x + c=$

$\displaystyle =\frac{3}{4}\sqrt[3]{x^4}+\sin x + c$

d) $\displaystyle \int \left(x^3+2x^2-1\right) \, \mathrm {d} x =\int x^3 \, \mathrm {d} x +2\int x^2 \, \mathrm {d} x-\int 1 \, \mathrm {d} x=$ 

$\displaystyle =  +\frac{2}{3}x^3-x + c\frac{1}{4}x^4$

e) $\displaystyle \int \left(e^x+\cos x\right) \, \mathrm {d} x = \int e^x \, \mathrm {d} x+ \int \cos x \, \mathrm {d} x=e^x + \sin x + c$

f) $\displaystyle \int \left(3^x+\frac{2}{x}\right) \, \mathrm {d} x= \int 3^x \, \mathrm {d} x+ 2\int \frac{1}{x} \, \mathrm {d} x=\frac{3^x }{\ln 3} + 2\ln \left | x \right | + c$

g) $\displaystyle \int \frac{3}{1+x^2} \, \mathrm {d} x =3\int \frac{1}{1+x^2} \, \mathrm {d} x =3\arctan x + c$