M171 Calculus I · Unit 4: Optimization & the integral · on Exam 4
§5.2 The second Fundamental Theorem of Calculus
Study notes for The second Fundamental Theorem of Calculus in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- Second FTC: $\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt = f(x)$. Differentiating an integral function returns the integrand evaluated at the upper limit.
- With a variable upper limit $g(x)$, the chain rule applies: $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\,g'(x)$.
- Every continuous function has an antiderivative, namely $\int_a^x f(t)dt$, even when no elementary formula exists (e.g. $\int_0^x e^{-t^2}dt$).
Key formulas
Second FTC
$$\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$$
With chain rule
$$\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f\big(g(x)\big)\,g'(x)$$
Worked example
Let $A(x) = \int_1^{x} (t^2 - 4)\,dt$. Find $A'(x)$, the critical numbers of $A$, and $\frac{d}{dx}\int_0^{x^2}\sin t\,dt$.
Show the solution
$A'(x) = x^2 - 4$, so $A' = 0$ at $x=\pm2$. $A$ has a local min at $x=2$ (where $A'$ goes $-$ to $+$) and a local max at $x=-2$. Last part: $\sin(x^2)\cdot 2x$.
Common mistakes
- Evaluating the integral first when the question only asks for the derivative.
- Missing the chain factor with a non-x upper limit.
Exam tipThe second FTC turns "differentiate an integral" into "copy the integrand." Do not integrate.
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