https://file.aiquickdraw.com/mathbot/file/1726688939383csn0b4jz.jpg

Let's solve the integrals step by step from the image.

### (a) \(\int_0^1 \frac{x^3 - 4x - 10}{x^2 - x - 6} \, dx\)

We first factor the denominator and perform partial fraction decomposition.

The denominator can be factored as:

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

Now, decompose the integrand:

\[
\frac{x^3 - 4x - 10}{(x - 3)(x + 2)} = \frac{A}{x - 3} + \frac{B}{x + 2} + \frac{Cx + D}{(x - 3)(x + 2)}
\]

We'll find constants \(A\), \(B\), \(C\), and \(D\) by equating coefficients. I'll calculate this next.

### (b) \(\int \arctan\left(\frac{1}{x}\right) \, dx\)

This is an integral of the arctangent function with a substitution:

Let \(u = \frac{1}{x}\), hence \(du = -\frac{1}{x^2} dx\).

The new integral becomes:

\[
-\int \frac{\arctan(u)}{u^2} \, du
\]

I'll proceed to calculate this as well.

### (c) \(\int_0^{\frac{\pi}{6}} \sec^3 \theta \tan \theta \, d\theta\)

Use a substitution here:

Let \(u = \sec \theta\), then \(du = \sec \theta \tan \theta d\theta\).

The integral transforms into:

\[
\int_1^{\sec \frac{\pi}{6}} u^2 \, du
\]

I'll now calculate all three integrals step by step. Let's proceed.Here are the results for the integrals:

### (a) \(\int_0^1 \frac{x^3 - 4x - 10}{x^2 - x - 6} \, dx\)
The result is:

\[
-\ln(2) + \ln(3) + \frac{3}{2} \approx 0.901
\]

### (b) \(\int \arctan\left(\frac{1}{x}\right) \, dx\)
The indefinite integral is:

\[
x \arctan\left(\frac{1}{x}\right) + \frac{1}{2} \ln(x^2 + 1)
\]

### (c) \(\int_0^{\frac{\pi}{6}} \sec^3 \theta \tan \theta \, d\theta\)
The result is:

\[
-\frac{1}{3} + \frac{8\sqrt{3}}{27} \approx 0.360
\]

Let me know if you'd like further details on the steps or the calculation process. 

Here are 5 questions to expand on this topic:

1. How does partial fraction decomposition work in integration?
2. Can you explain the substitution method in integral (b)?
3. How do trigonometric identities help simplify integral (c)?
4. What are some common techniques to handle improper integrals?
5. How can logarithmic terms appear in definite integrals?

**Tip:** When working with logarithmic integrals, always check if the argument of the logarithm simplifies to avoid undefined values during evaluation.

Evaluate the following integrals: a) ∫ (x^3 - 4x - 10) / (x^2 - x - 6) dx from 0 to 1, b) ∫ arctan(1/x) dx, c) ∫ sec^3(θ) tan(θ) dθ from 0 to π/6.

Learn how to evaluate definite integrals such as ∫ (x^3 - 4x - 10) / (x^2 - x - 6) dx, involving partial fraction decomposition, and integrals with trigonometric substitution. Detailed solutions for Grades 11-12.

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