The image contains a set of integrals to be solved using the method of partial fractions. Let's go through one of them step by step.

Problem (a):

$\int \frac{-8x^3 + 82x^2 - 45x - 30}{15x^2(x - 2)(x + 1)} \, dx$

We will decompose this rational function using partial fractions. The strategy is to express the integrand as a sum of simpler fractions.

Step 1: Set up the partial fraction decomposition

The denominator is $15x^2(x - 2)(x + 1)$, which can be decomposed into three terms:

• A term for $x^2$,
• A term for $x - 2$,
• A term for $x + 1$.

Thus, the partial fraction decomposition will look like: $\frac{-8x^3 + 82x^2 - 45x - 30}{15x^2(x - 2)(x + 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 2} + \frac{D}{x + 1}$

Step 2: Solve for the constants $A$, $B$, $C$, and $D$

We now multiply both sides by the denominator $15x^2(x - 2)(x + 1)$ to get rid of the fractions: $-8x^3 + 82x^2 - 45x - 30 = A \cdot 15x(x - 2)(x + 1) + B \cdot 15(x - 2)(x + 1) + C \cdot 15x^2(x + 1) + D \cdot 15x^2(x - 2)$

Next, expand both sides and compare the coefficients of powers of $x$ to solve for $A$, $B$, $C$, and $D$.

Would you like me to proceed with solving for these constants, or would you like help with another part of the problem?

Follow-up Questions:

1. Would you like a more detailed breakdown of the steps involved in partial fraction decomposition?
2. Do you need assistance with one of the other integrals (b, c, or d)?
3. Are there any specific areas within partial fractions that you find challenging?
4. Would you like an explanation of how to handle repeated roots or irreducible quadratics in partial fractions?
5. Do you prefer a focus on the integration process after the decomposition?

Tip:

When solving for constants in partial fractions, it's often helpful to substitute specific values of $x$ to simplify the system of equations.