Comprehensive list of solved indefinite integrals covering power rules, logarithms, and trigonometric functions.

Let’s solve this step by step.

We are given a list of functions labeled from a) to Z), and we’re expected to match each function with its correct antiderivative (indefinite integral). But wait — actually, looking at the structure, it seems like these are all *answers* (antiderivatives), and perhaps the original problem was to find which derivative matches which? Or maybe it's a matching exercise where each letter corresponds to an integral result?

But since no question is explicitly stated in your message, and you said “solve the problem accurately”, I need to infer what the task is.

Looking again: The image shows 26 expressions (from a to Z), each being an antiderivative (they all have “+ c”). This suggests that the original homework problem was likely:

> “Match each integral below with its correct antiderivative from the list.”

But since the integrals themselves are not shown here — only the answers — there must be missing context.

Wait — perhaps this is a standard matching worksheet where students are given derivatives or integrands on one side and antiderivatives on the other, and they must pair them.

Since you uploaded only the answer key part (the antiderivatives), and didn’t provide the questions (the integrands or derivatives), I cannot directly “solve” anything unless I assume something.

Alternatively — maybe the task is to verify which of these are correctly written antiderivatives? But that doesn’t make sense without knowing what they’re supposed to be antiderivatives of.

Another possibility: Perhaps this is a multiple-choice style matching game, and the user expects me to recognize common patterns and group them logically? But again, without the left-hand side (the functions to integrate), I can't proceed meaningfully.

Hold on — let me re-read your instruction:

> You are an educational assistant helping a student solve homework problems.
> The user uploaded an image containing a task.
> Solve the problem accurately.

And then you listed options: ExplainAnswer: false, TeachTopic: false, GeneratePractice: false.

So according to your rules, if all those are false, I should just give the Final Answer — but only after working through the problem.

But here’s the issue: There is no actual problem stated. Only a list of antiderivatives.

Unless... perhaps the real task is hidden in the formatting? Let me look again.

Actually, upon closer inspection — this looks exactly like a standard calculus matching worksheet where Column A has integrals and Column B has antiderivatives, and students draw lines between them. Since only Column B is visible here (labeled a to Z), and no Column A is provided, I cannot complete the matching.

This means either:
1. The image was cropped or incomplete.
2. The user forgot to include the integrands.
3. The task is different — e.g., simplify or classify these expressions.

Given that, and since I must follow instructions strictly, I will assume that the intended task is to identify which of these antiderivatives correspond to basic integration formulas, and perhaps point out any errors — but none appear obviously wrong at first glance.

Alternatively — maybe the task is to compute the derivative of each expression to see what function it came from? That could work!

Yes! That makes sense. If these are antiderivatives, then taking their derivative should give us back the original integrand. So perhaps the homework was: “For each of the following antiderivatives, find the corresponding integrand by differentiation.”

That would be a valid exercise — testing understanding of the relationship between derivatives and antiderivatives.

Let’s try that approach.

I’ll pick a few examples to demonstrate, then generalize.

Take option a):
F(x) = (1/3)√(5 + 2x)³ + c
= (1/3)(5 + 2x)^{3/2} + c

Differentiate using chain rule:

F’(x) = (1/3) * (3/2)(5 + 2x)^{1/2} * 2
= (1/3)*(3/2)*2 * √(5 + 2x)
= 1 * √(5 + 2x)

So the integrand would be √(5 + 2x)

Similarly, take b):
F(x) = [(3x² - 4)^6]/36 + c

F’(x) = [6*(3x² - 4)^5 * 6x] / 36
Wait — derivative of inside is 6x? No:

d/dx (3x² - 4) = 6x? Actually, 3x² → derivative is 6x? Wait no: d/dx(3x²) = 6x? Yes.

Wait: 3x² derivative is 6x? Actually, no: d/dx(3x²) = 6x? Let me calculate:

d/dx (3x²) = 3 * 2x = 6x — yes.

So:

F’(x) = [6 * (3x² - 4)^5 * 6x] / 36 ? No — wait:

Chain rule: d/dx [u^n] = n u^{n-1} * u'

Here u = 3x² - 4, n = 6

So F’(x) = [6 * (3x² - 4)^5 * (6x)] / 36

Simplify: 6 * 6x = 36x, divided by 36 → x

So F’(x) = x * (3x² - 4)^5

So integrand is x(3x² - 4)^5

Okay, so this method works.

But doing this for all 26 items would be very long — and since the user didn’t specify which ones to do, or what the goal is, I’m stuck.

Perhaps the real task is simpler: Maybe the student is supposed to recognize that each of these is a correct antiderivative form, and the "problem" is just to acknowledge that? But that doesn’t require solving.

Another idea: In some worksheets, letters are matched to numbers, and the student fills in blanks. But again, without the other column, impossible.

Given the constraints, and since all three options (ExplainAnswer, TeachTopic, GeneratePractice) are set to false, I am required to output ONLY the Final Answer — but only after reasoning.

But what is the final answer to what?

I think there might be a misunderstanding. Perhaps the user intended to upload both sides of the matching sheet, but only uploaded the answer side.

In that case, the most reasonable thing I can do is to state that the problem cannot be solved as presented, because the integrands (or derivatives) are missing.

However, since I must follow the format, and provide a Final Answer after reasoning, I will conclude with:

The list provided contains 26 antiderivatives (indefinite integrals) labeled from a to Z. Without the corresponding integrands or derivatives to match them to, no specific solution can be determined. Each expression is mathematically valid as an antiderivative of some function, obtainable via reverse differentiation.

Final Answer: Cannot be determined without the corresponding integrands or derivatives to match.