Evaluate the following integrals: 1. integral (2x+3)/(x2-3x+2) dx 2a. integral (x7 + 4x6 - 2x5 + (1)/(x) + e^x + x) dx

Here are the solutions to the integration problems:

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

Step 1: Factor the denominator and decompose the integrand into partial fractions.
The denominator $x^2-3x+2$ factors as $(x-1)(x-2)$.
We set up the partial fraction decomposition:

$$\frac{2x+3}{(x-1)(x-2)} = \frac{A}{x-1} + \frac{B}{x-2}$$

Multiplying both sides by $(x-1)(x-2)$ gives:

$$2x+3 = A(x-2) + B(x-1)$$

To find $A$, set $x=1$:

$$2(1)+3 = A(1-2) + B(1-1) \implies 5 = -A \implies A = -5$$

To find $B$, set $x=2$:

$$2(2)+3 = A(2-2) + B(2-1) \implies 7 = B$$

So, the integral becomes:

$$\int \left( \frac{-5}{x-1} + \frac{7}{x-2} \right) dx$$

Step 2: Integrate each term.

$$\int \frac{-5}{x-1} dx = -5 \ln|x-1|$$ $$\int \frac{7}{x-2} dx = 7 \ln|x-2|$$

Step 3: Combine the results and add the constant of integration.

$$\int \frac{2x+3}{x^2-3x+2} dx = -5 \ln|x-1| + 7 \ln|x-2| + C$$

The final answer is $\boxed{-5 \ln|x-1| + 7 \ln|x-2| + C}$.

2. Evaluate the following: