How do you verify that a proposed function is an antiderivative of a given integrand?
Conditions
- The proposed function is differentiable.
- The original integrand is defined on the domain of interest.
Reasoning, step by step
- Identify the proposed antiderivative .
- Identify the original integrand .
- Compute the derivative .
- Compare with .
- If for all in the domain, the verification is successful.
- If they differ, the proposed function is not an antiderivative of .
Example
The script states: 'Differentiating the proposed result should recover the original integrand.' For instance, differentiating yields , verifying it as the antiderivative of .
Common misconceptions
- Integrating the proposed function instead of differentiating it.
- Forgetting that the constant disappears during differentiation, so its presence or absence in the check doesn't invalidate the core derivative match.
- Checking only at a single point rather than generally.
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