Evaluate the following integrals:
(i) `int_2^3 x^2 dx`
(ii) `int_1^3 x/((x+1)(x+2)) dx`

Answer

Let's evaluate the given integrals step by step.

### (i) Evaluate the integral \( \int_2^3 x^2 \, dx \)

1. **Find the antiderivative**:
The antiderivative of \( x^2 \) is \( \frac{x^3}{3} \).

2. **Apply the limits**:
We will evaluate the antiderivative from 2 to 3:
\[ \left[ \frac{x^3}{3} \right]_2^3 = \frac{3^3}{3} - \frac{2^3}{3} \]

3. **Calculate the values**:
\[ = \frac{27}{3} - \frac{8}{3} = \frac{27 - 8}{3} = \frac{19}{3} \]

Thus, the value of the integral \( \int_2^3 x^2 \, dx = \frac{19}{3} \).

### (ii) Evaluate the integral \( \int_1^3 \frac{x}{(x+1)(x+2)} \, dx \)

1. **Partial Fraction Decomposition**:
We can express \( \frac{x}{(x+1)(x+2)} \) as:
\[ \frac{x}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2} \]
Multiplying through by the denominator \( (x+1)(x+2) \) gives:
\[ x = A(x+2) + B(x+1) \]
Expanding the right side:
\[ x = Ax + 2A + Bx + B = (A + B)x + (2A + B) \]
Setting coefficients equal:
- For \( x \): \( A + B = 1 \)
- For the constant term: \( 2A + B = 0 \)

2. **Solve the system of equations**:
From \( A + B = 1 \), we can express \( B = 1 - A \).
Substituting into the second equation:
\[ 2A + (1 - A) = 0 \implies A + 1 = 0 \implies A = -1 \]
Then substituting back to find \( B \):
\[ B = 1 - (-1) = 2 \]
Thus, we have:
\[ \frac{x}{(x+1)(x+2)} = \frac{-1}{x+1} + \frac{2}{x+2} \]

3. **Rewrite the integral**:
\[ \int_1^3 \left( \frac{-1}{x+1} + \frac{2}{x+2} \right) \, dx \]
This can be split into two integrals:
\[ = -\int_1^3 \frac{1}{x+1} \, dx + 2\int_1^3 \frac{1}{x+2} \, dx \]

4. **Evaluate the integrals**:
- The integral \( \int \frac{1}{x+1} \, dx = \ln|x+1| \)
- The integral \( \int \frac{1}{x+2} \, dx = \ln|x+2| \)

Therefore:
\[ -\left[ \ln|x+1| \right]_1^3 + 2\left[ \ln|x+2| \right]_1^3 \]

5. **Calculate each part**:
- For \( -\left[ \ln|x+1| \right]_1^3 \):
\[ -(\ln(4) - \ln(2)) = -\ln(4/2) = -\ln(2) \]
- For \( 2\left[ \ln|x+2| \right]_1^3 \):
\[ 2(\ln(5) - \ln(3)) = 2\ln(5/3) \]

6. **Combine the results**:
\[ -\ln(2) + 2\ln(5/3) = -\ln(2) + \ln((5/3)^2) = -\ln(2) + \ln(25/9) \]
\[ = \ln\left(\frac{25/9}{2}\right) = \ln\left(\frac{25}{18}\right) \]

Thus, the value of the integral \( \int_1^3 \frac{x}{(x+1)(x+2)} \, dx = \ln\left(\frac{25}{18}\right) \).

### Summary of Results
1. \( \int_2^3 x^2 \, dx = \frac{19}{3} \)
2. \( \int_1^3 \frac{x}{(x+1)(x+2)} \, dx = \ln\left(\frac{25}{18}\right) \)