The power rule formula ∫xⁿ dx = xⁿ⁺¹/(n+1) + C fails when n = -1 because the denominator becomes zero — division by zero is undefined. The correct antiderivative is ln|x| + C, because d/dx[ln|x|] = 1/x. The absolute value is necessary to extend the domain to negative x, where ln would otherwise be undefined.
Question 2 Multiple Choice
A student writes ∫sin(x) dx = cos(x) + C. What is wrong with this answer?
ANothing — both cos(x) and -cos(x) are valid antiderivatives of sin(x)
BThe sign is wrong; the correct answer is -cos(x) + C
CTrigonometric functions cannot be integrated using basic rules
DThe answer should be -sin(x) + C
Since d/dx[cos(x)] = -sin(x), running this backwards gives ∫sin(x) dx = -cos(x) + C, not +cos(x). You can verify: differentiate -cos(x) to get -(-sin(x)) = sin(x). The minus sign is not an error — it reflects the minus sign already present in the derivative of cosine. This is the most common sign mistake in basic integration.
Question 3 True / False
The integral of eˣ dx is eˣ + C.
TTrue
FFalse
Answer: True
Since d/dx[eˣ] = eˣ, running this rule in reverse gives ∫eˣ dx = eˣ + C. The exponential function is its own derivative, making it also its own antiderivative. This is one of the simplest integrals precisely because no sign change or exponent adjustment occurs.
Question 4 True / False
There is a product rule for integration analogous to the product rule for differentiation: ∫f(x)g(x) dx = (∫f dx)(∫g dx).
TTrue
FFalse
Answer: False
No such product rule exists for integration. Unlike differentiation, integration has no formula that lets you integrate a product by integrating each factor separately. Integration by parts exists for integrating products, but it is not a simple multiplicative rule — it requires a specific setup and transforms the problem rather than resolving it directly. Assuming a product rule is a common and costly error.
Question 5 Short Answer
Explain why differentiating your answer is a reliable way to check an integration result, and use this method to find the correct sign of ∫sin(x) dx.
Think about your answer, then reveal below.
Model answer: An integral asks for a function whose derivative is the integrand. Differentiating the answer directly checks this. For ∫sin(x) dx, try -cos(x): d/dx[-cos(x)] = -(-sin(x)) = sin(x). Since we recover the original integrand, -cos(x) + C is correct.
This verification habit works because differentiation and integration are inverse operations. The derivative of the answer must equal the integrand — if it doesn't, the answer is wrong. For trig integrals especially, where sign errors are common, this check catches mistakes immediately rather than propagating them through a longer problem.