When an Antiderivative Is Supplied
A definite integral can sometimes be evaluated without first finding an antiderivative: the problem gives you a function that is claimed to be one. The essential first step is to verify that claim. A function that looks plausible, or that gives convenient endpoint values, is not enough. Its derivative must match the integrand on the interval of integration.
Once that match is established and the conditions for the Fundamental Theorem of Calculus are satisfied, evaluate the supplied function at the endpoints and subtract in the correct order. This tutorial focuses on that sequence: check the derivative, then calculate the endpoint difference.
The Conditions and the Evaluation Rule
For a definite integral over \([a,b]\), the standard Fundamental Theorem of Calculus evaluation rule applies when \(f\) is continuous on the closed interval \([a,b]\), and \(F\) is an antiderivative of \(f\) on the interval. In particular, \(F'(x)=f(x)\) for interior points, and \(F\) must have defined endpoint values \(F(a)\) and \(F(b)\).
The condition that \(f\) be continuous on the interval is a familiar sufficient condition for this version of the theorem. Do not assume the formula is justified merely because endpoint values can be calculated: the derivative match and the theorem's conditions matter. When the integrand or proposed antiderivative is piecewise-defined, check the relevant pieces and interval conditions rather than treating the formula as one smooth expression everywhere.
The order of subtraction is part of the rule: upper endpoint first, lower endpoint second. The notation \(F(x)\big|_a^b\) means \(F(b)-F(a)\), not the reverse.
Why does checking the derivative come before substituting endpoints? The Fundamental Theorem connects the accumulated integral of \(f\) to the change in a function whose derivative is \(f\). If the supplied function has some other derivative, its endpoint change describes accumulation of that other function, not necessarily accumulation of \(f\). The derivative check identifies whether the supplied function is the one the theorem requires.
A Reliable Evaluation Process
Identify \(a\), \(b\), and the function \(f(x)\) inside the integral. Keep the variable \(x\) consistent while differentiating and evaluating.
Compute \(F'(x)\) using the relevant differentiation rules. Do not substitute endpoints yet.
Confirm that \(F'(x)=f(x)\) throughout the interval, and check the conditions for the Fundamental Theorem.
Calculate \(F(b)\) and \(F(a)\) separately, then form \(F(b)-F(a)\).
Revisit the endpoint substitutions and confirm that the upper-endpoint value was placed first.
Worked Examples
Worked Example: A Polynomial Integrand
Evaluate \(\int_{-1}^{2}(3x^2-4x+1)\,dx\), given the proposed antiderivative \(F(x)=x^3-2x^2+x\).
First differentiate each term:
This equals the integrand for every \(x\). Both the integrand and \(F\) are polynomials, so they are continuous on \([-1,2]\), and the Fundamental Theorem applies.
Now substitute the upper endpoint:
Substitute the lower endpoint, retaining the parentheses around \(-1\):
Subtract the lower value from the upper value:
Check: the endpoint values are \(2\) and \(-4\), so their change from the lower endpoint to the upper endpoint is \(2-(-4)=6\), consistent with the stated evaluation.
Worked Example: A Trigonometric Antiderivative
Evaluate \(\int_0^{\pi/2}2\sin x\cos x\,dx\), given \(F(x)=\sin^2x\).
Use the chain rule to verify the derivative. The outer function \(u^2\) has derivative \(2u\), and the inner function \(\sin x\) has derivative \(\cos x\):
The derivative matches the integrand. The integrand and \(F\) are continuous on \([0,\pi/2]\), so the theorem applies.
Evaluate the upper value using \(\sin(\pi/2)=1\):
Evaluate the lower value using \(\sin(0)=0\):
Therefore,
Check: the verified derivative is exactly \(2\sin x\cos x\), and the endpoint difference is \(1\), with no approximation required.
Worked Example: A Logarithmic Antiderivative
Evaluate \(\int_1^e \frac{1+\ln x}{x}\,dx\), given \(F(x)=\ln x+\frac{1}{2}(\ln x)^2\).
The interval \([1,e]\) lies within the domain \(x>0\) of \(\ln x\). Differentiate both terms, using \(\frac{d}{dx}\ln x=\frac1x\) and the chain rule:
This is the integrand. It is continuous on \([1,e]\), so the evaluation rule applies.
Since \(\ln e=1\), the upper-endpoint value is
Since \(\ln 1=0\), the lower-endpoint value is
Thus,
Check: differentiating the supplied expression reproduces the full numerator and denominator of the integrand, and the exact endpoint values give \(\frac{3}{2}\).
Common Mistake: Substituting Before Verifying
Suppose you need to evaluate \(\int_0^2(2x+1)\,dx\), and someone supplies \(G(x)=x^2+2x\) as an antiderivative. A tempting but invalid calculation is:
The endpoint arithmetic is correct, but the result does not follow from the Fundamental Theorem for the stated integrand. Differentiate first:
Thus \(G\) is not an antiderivative of the integrand. The calculation \(G(2)-G(0)=8\) cannot be used to evaluate the given integral. If the correct antiderivative \(F(x)=x^2+x\) is supplied instead, verify it and then evaluate:
The correction is not to adjust the endpoint arithmetic until it produces a plausible answer. It is to reject the mismatched candidate and use a function whose derivative actually equals the integrand.
Why the Check Is Useful
A supplied expression can contain a small but consequential error: a missing coefficient, a sign change, or a missing factor from the chain rule. These errors can be hard to notice by looking at the expression alone. Differentiation provides a direct test, and performing it before endpoint substitution prevents an invalid result from being hidden by otherwise correct arithmetic.
For a practical self-check, keep the two stages visibly separate in your work. First write a line showing \(F'(x)=f(x)\). Then, only after confirming the equality and the theorem's conditions, write \(F(b)-F(a)\). This makes it easier to locate an error: a mismatch belongs to the derivative check, while an incorrect result after a valid match usually belongs to endpoint substitution or subtraction.
Check Your Understanding
For each question, state the derivative check or endpoint calculation that supports your answer.
- For \(\int_0^1(6x-2)\,dx\) with supplied \(F(x)=3x^2-2x\), what derivative should you compute, and what endpoint difference results?
- A proposed antiderivative for \(f(x)=\cos x\) is \(G(x)=\sin x\). What derivative check confirms or rejects the proposal?
- For \(\int_1^2 \frac{1}{x}\,dx\) with supplied \(F(x)=\ln x\), evaluate \(F(2)-F(1)\) exactly.
- What is the error in using \(H(b)-H(a)\) when \(H'(x)\ne f(x)\), even if both endpoint values are easy to calculate?
- In the evaluation rule, which endpoint value is subtracted, and how can you write the rule using \(F(x)\big|_a^b\)?
