The problem asks us to find the antiderivatives (indefinite integrals) of a set of given functions. We need to find the antiderivative for each function listed from a) to r).

1. Problem Description

The problem asks us to find the antiderivatives (indefinite integrals) of a set of given functions. We need to find the antiderivative for each function listed from a) to r).

2. Solution Steps

a) Find the antiderivative of $x^7$.

Using the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, we have

$\int x^7 dx = \frac{x^{7+1}}{7+1} + C = \frac{x^8}{8} + C$.

b) Find the antiderivative of $\sqrt[3]{x}$.

First, rewrite the expression as $x^{1/3}$. Then, using the power rule for integration:

$\int x^{1/3} dx = \frac{x^{(1/3)+1}}{(1/3)+1} + C = \frac{x^{4/3}}{4/3} + C = \frac{3}{4}x^{4/3} + C = \frac{3}{4}\sqrt[3]{x^4} + C$.

c) Find the antiderivative of $\frac{1}{x^3}$.

Rewrite the expression as $x^{-3}$. Then, using the power rule for integration:

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

d) Find the antiderivative of $\frac{1}{\sqrt{x}}$.

Rewrite the expression as $x^{-1/2}$. Then, using the power rule for integration:

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

e) Find the antiderivative of $7x$.

Using the power rule for integration:

$\int 7x dx = 7 \int x^1 dx = 7 \cdot \frac{x^{1+1}}{1+1} + C = 7 \cdot \frac{x^2}{2} + C = \frac{7}{2}x^2 + C$.

f) Find the antiderivative of $\ln 2$.

Since $\ln 2$ is a constant, the antiderivative is:

$\int \ln 2 dx = (\ln 2) \int 1 dx = (\ln 2)x + C = x\ln 2 + C$.

g) Find the antiderivative of $\frac{e^3}{x}$.

Since $e^3$ is a constant, the antiderivative is:

$\int \frac{e^3}{x} dx = e^3 \int \frac{1}{x} dx = e^3 \ln|x| + C$.

h) Find the antiderivative of $x\ln 3$.

Since $\ln 3$ is a constant, the antiderivative is:

$\int x\ln 3 dx = \ln 3 \int x dx = \ln 3 \cdot \frac{x^2}{2} + C = \frac{x^2}{2}\ln 3 + C$.

i) Find the antiderivative of $\frac{1}{x\ln 2}$.

Since $\ln 2$ is a constant, the antiderivative is:

$\int \frac{1}{x\ln 2} dx = \frac{1}{\ln 2} \int \frac{1}{x} dx = \frac{1}{\ln 2} \ln|x| + C = \frac{\ln|x|}{\ln 2} + C$.

j) Find the antiderivative of $3x + \frac{1}{3x}$.

$\int (3x + \frac{1}{3x}) dx = \int 3x dx + \int \frac{1}{3x} dx = 3\int x dx + \frac{1}{3}\int \frac{1}{x} dx = 3 \cdot \frac{x^2}{2} + \frac{1}{3} \ln|x| + C = \frac{3}{2}x^2 + \frac{1}{3}\ln|x| + C$.

k) Find the antiderivative of $\frac{e}{x} + \frac{x}{e}$.

$\int (\frac{e}{x} + \frac{x}{e}) dx = \int \frac{e}{x} dx + \int \frac{x}{e} dx = e\int \frac{1}{x} dx + \frac{1}{e}\int x dx = e\ln|x| + \frac{1}{e} \cdot \frac{x^2}{2} + C = e\ln|x| + \frac{x^2}{2e} + C$.

l) Find the antiderivative of $xe^{-2} + ex^{-1}$.

Since $e^{-2}$ is a constant, and also $e$ is a constant, the antiderivative is:

$\int (xe^{-2} + ex^{-1}) dx = e^{-2}\int x dx + e\int \frac{1}{x} dx = e^{-2}\frac{x^2}{2} + e\ln|x| + C = \frac{x^2}{2e^2} + e\ln|x| + C$.

m) Find the antiderivative of $(e^2 - 2^e)e^x$.

Since $e^2 - 2^e$ is a constant, the antiderivative is:

$\int (e^2 - 2^e)e^x dx = (e^2 - 2^e) \int e^x dx = (e^2 - 2^e) e^x + C$.

n) Find the antiderivative of $\sqrt{3x}$.

$\int \sqrt{3x} dx = \int \sqrt{3}\sqrt{x} dx = \sqrt{3}\int x^{1/2} dx = \sqrt{3} \frac{x^{(1/2)+1}}{(1/2)+1} + C = \sqrt{3} \frac{x^{3/2}}{3/2} + C = \sqrt{3} \cdot \frac{2}{3}x^{3/2} + C = \frac{2\sqrt{3}}{3}x^{3/2} + C = \frac{2\sqrt{3}}{3} x\sqrt{x} + C$.

o) Find the antiderivative of $ex^{e+1}$.

$\int ex^{e+1} dx = e \int x^{e+1} dx = e \frac{x^{(e+1)+1}}{(e+1)+1} + C = e \frac{x^{e+2}}{e+2} + C = \frac{ex^{e+2}}{e+2} + C$.

p) Find the antiderivative of $x^7 + 7x + \frac{7}{x} + 7$.

$\int (x^7 + 7x + \frac{7}{x} + 7) dx = \int x^7 dx + \int 7x dx + \int \frac{7}{x} dx + \int 7 dx = \frac{x^8}{8} + \frac{7x^2}{2} + 7\ln|x| + 7x + C$.

q) Find the antiderivative of $e^x + x^e + e + x$.

$\int (e^x + x^e + e + x) dx = \int e^x dx + \int x^e dx + \int e dx + \int x dx = e^x + \frac{x^{e+1}}{e+1} + ex + \frac{x^2}{2} + C$.

r) Find the antiderivative of $7x^2 - 3x + 8 + \frac{1}{x} + \frac{2}{x^2}$.

$\int (7x^2 - 3x + 8 + \frac{1}{x} + \frac{2}{x^2}) dx = \int 7x^2 dx - \int 3x dx + \int 8 dx + \int \frac{1}{x} dx + \int \frac{2}{x^2} dx = 7\int x^2 dx - 3\int x dx + 8\int 1 dx + \int \frac{1}{x} dx + 2\int x^{-2} dx = 7\cdot \frac{x^3}{3} - 3\cdot \frac{x^2}{2} + 8x + \ln|x| + 2\cdot \frac{x^{-1}}{-1} + C = \frac{7}{3}x^3 - \frac{3}{2}x^2 + 8x + \ln|x| - \frac{2}{x} + C$.

3. Final Answer

a) $\frac{x^8}{8} + C$

b) $\frac{3}{4}x^{4/3} + C$

c) $-\frac{1}{2x^2} + C$

d) $2\sqrt{x} + C$

e) $\frac{7}{2}x^2 + C$

f) $x\ln 2 + C$

g) $e^3\ln|x| + C$

h) $\frac{x^2}{2}\ln 3 + C$

i) $\frac{\ln|x|}{\ln 2} + C$

j) $\frac{3}{2}x^2 + \frac{1}{3}\ln|x| + C$

k) $e\ln|x| + \frac{x^2}{2e} + C$

l) $\frac{x^2}{2e^2} + e\ln|x| + C$

m) $(e^2 - 2^e) e^x + C$

n) $\frac{2\sqrt{3}}{3} x\sqrt{x} + C$

o) $\frac{ex^{e+2}}{e+2} + C$

p) $\frac{x^8}{8} + \frac{7x^2}{2} + 7\ln|x| + 7x + C$

q) $e^x + \frac{x^{e+1}}{e+1} + ex + \frac{x^2}{2} + C$

r) $\frac{7}{3}x^3 - \frac{3}{2}x^2 + 8x + \ln|x| - \frac{2}{x} + C$