Integrate the following expressions: a) x2 b) 4x5 c) 2√x d) 7 e) x3+x2+x+1

$$\int x^2 \,dx = \frac{x^{2+1}}{2+1} + C$$ $$= \frac{x^3}{3} + C$$

The integral is $\boxed{\frac{x^3}{3} + C}$.

$$\int 4x^5 \,dx = 4 \int x^5 \,dx$$ $$= 4 \left(\frac{x^{5+1}}{5+1}\right) + C$$ $$= 4 \left(\frac{x^6}{6}\right) + C$$ $$= \frac{2x^6}{3} + C$$

The integral is $\boxed{\frac{2x^6}{3} + C}$.

$$\int 2x^{1/2} \,dx = 2 \int x^{1/2} \,dx$$ $$= 2 \left(\frac{x^{1/2+1}}{1/2+1}\right) + C$$ $$= 2 \left(\frac{x^{3/2}}{3/2}\right) + C$$ $$= 2 \cdot \frac{2}{3} x^{3/2} + C$$ $$= \frac{4}{3} x^{3/2} + C$$

The integral is $\boxed{\frac{4}{3} x^{3/2} + C}$.

$$\int 7 \,dx = 7x + C$$

The integral is $\boxed{7x + C}$.

$$\int (x^3+x^2+x+1) \,dx = \int x^3 \,dx + \int x^2 \,dx + \int x \,dx + \int 1 \,dx$$ $$= \frac{x^{3+1}}{3+1} + \frac{x^{2+1}}{2+1} + \frac{x^{1+1}}{1+1} + x + C$$ $$= \frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} + x + C$$

The integral is $\boxed{\frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} + x + C}$.

$$\int x^{3/5} \,dx = \frac{x^{3/5+1}}{3/5+1} + C$$ $$= \frac{x^{8/5}}{8/5} + C$$ $$= \frac{5}{8} x^{8/5} + C$$

The integral is $\boxed{\frac{5}{8} x^{8/5} + C}$.

$$\int \frac{x^6}{2} \,dx = \frac{1}{2} \int x^6 \,dx$$ $$= \frac{1}{2} \left(\frac{x^{6+1}}{6+1}\right) + C$$ $$= \frac{1}{2} \left(\frac{x^7}{7}\right) + C$$ $$= \frac{x^7}{14} + C$$

The integral is $\boxed{\frac{x^7}{14} + C}$.

j) Integrate $x-\sqrt{x}$.

$$\int (x-x^{1/2}) \,dx = \int x \,dx - \int x^{1/2} \,dx$$ $$= \frac{x^{1+1}}{1+1} - \frac{x^{1/2+1}}{1/2+1} + C$$ $$= \frac{x^2}{2} - \frac{x^{3/2}}{3/2} + C$$ $$= \frac{x^2}{2} - \frac{2}{3} x^{3/2} + C$$

The integral is $\boxed{\frac{x^2}{2} - \frac{2}{3} x^{3/2} + C}$.

k) Integrate $\frac{(x+3)^2}{2x^4}$.

$$(x+3)^2 = x^2 + 6x + 9$$ $$\frac{(x+3)^2}{2x^4} = \frac{x^2 + 6x + 9}{2x^4}$$ $$= \frac{x^2}{2x^4} + \frac{6x}{2x^4} + \frac{9}{2x^4}$$ $$= \frac{1}{2}x^{-2} + 3x^{-3} + \frac{9}{2}x^{-4}$$

$$\int \left(\frac{1}{2}x^{-2} + 3x^{-3} + \frac{9}{2}x^{-4}\right) \,dx$$ $$= \frac{1}{2} \int x^{-2} \,dx + 3 \int x^{-3} \,dx + \frac{9}{2} \int x^{-4} \,dx$$ $$= \frac{1}{2} \left(\frac{x^{-2+1}}{-2+1}\right) + 3 \left(\frac{x^{-3+1}}{-3+1}\right) + \frac{9}{2} \left(\frac{x^{-4+1}}{-4+1}\right) + C$$ $$= \frac{1}{2} \left(\frac{x^{-1}}{-1}\right) + 3 \left(\frac{x^{-2}}{-2}\right) + \frac{9}{2} \left(\frac{x^{-3}}{-3}\right) + C$$ $$= -\frac{1}{2}x^{-1} - \frac{3}{2}x^{-2} - \frac{3}{2}x^{-3} + C$$

$$= -\frac{1}{2x} - \frac{3}{2x^2} - \frac{3}{2x^3} + C$$

The integral is $\boxed{-\frac{1}{2x} - \frac{3}{2x^2} - \frac{3}{2x^3} + C}$.