Let us solve the integrals step by step.

Problem 1:

$\int \frac{2}{3e^x} \, dx$

Simplify: $\int \frac{2}{3} e^{-x} \, dx = \frac{2}{3} \int e^{-x} \, dx$

The integral of $e^{-x}$ is $-e^{-x}$: $\frac{2}{3} (-e^{-x}) + C = -\frac{2}{3} e^{-x} + C$

Problem 2:

$\int \frac{3x^2 - 5x}{x} \, dx$

Simplify the fraction: $\int \left( 3x - 5 \right) \, dx$

Integrate each term: $\int 3x \, dx - \int 5 \, dx = \frac{3x^2}{2} - 5x + C$

Problem 3:

$\int \frac{3}{7 \, \sqrt[5]{x^4}} \, dx$

Rewrite $\sqrt[5]{x^4} = x^{4/5}$: $\int \frac{3}{7} x^{-4/5} \, dx = \frac{3}{7} \int x^{-4/5} \, dx$

The integral of $x^n$ is $\frac{x^{n+1}}{n+1}$: $\frac{3}{7} \cdot \frac{x^{-4/5 + 1}}{-4/5 + 1} = \frac{3}{7} \cdot \frac{x^{1/5}}{1/5} = \frac{3}{7} \cdot 5x^{1/5} = \frac{15}{7} x^{1/5} + C$

Problem 4:

$\int \frac{(2 + t)^2}{\sqrt{t}} \, dt$

Expand $(2 + t)^2 = 4 + 4t + t^2$: $\int \frac{4 + 4t + t^2}{\sqrt{t}} \, dt = \int \frac{4}{\sqrt{t}} \, dt + \int \frac{4t}{\sqrt{t}} \, dt + \int \frac{t^2}{\sqrt{t}} \, dt$

Simplify each term:

1. $\int \frac{4}{\sqrt{t}} \, dt = 4 \int t^{-1/2} \, dt = 4 \cdot 2t^{1/2} = 8\sqrt{t}$
2. $\int \frac{4t}{\sqrt{t}} \, dt = 4 \int t^{1/2} \, dt = 4 \cdot \frac{2}{3} t^{3/2} = \frac{8}{3} t^{3/2}$
3. $\int \frac{t^2}{\sqrt{t}} \, dt = \int t^{3/2} \, dt = \frac{2}{5} t^{5/2}$

Combine: $8\sqrt{t} + \frac{8}{3} t^{3/2} + \frac{2}{5} t^{5/2} + C$

Final Answers:

1. $-\frac{2}{3} e^{-x} + C$
2. $\frac{3x^2}{2} - 5x + C$
3. $\frac{15}{7} x^{1/5} + C$
4. $8\sqrt{t} + \frac{8}{3} t^{3/2} + \frac{2}{5} t^{5/2} + C$

Would you like detailed steps for any specific part?

5 Related Questions:

1. What is the general formula for integrating exponential functions?
2. How does simplifying the integrand help in solving complex integrals?
3. Can the power rule for integration be applied to fractional exponents?
4. How is the expansion of polynomials useful in integration?
5. What are some common mistakes to avoid when integrating expressions?

Tip:

Always simplify the integrand before starting to integrate—it can save time and reduce the chance of errors!