Term-by-term integration is the technique of integrating a power series one term at a time, treating each term like an ordinary polynomial term, to produce a new power series for the antiderivative. On AP Calc BC it's a core move in Unit 10 for building new Taylor and Maclaurin series from known ones.
Term-by-term integration is exactly what it sounds like. If a function is written as a power series, you can find a series for its antiderivative by integrating each term individually, just like you'd integrate a polynomial. The series for f(x) = 1 + x + x² + x³ + ... integrates to C + x + x²/2 + x³/3 + x⁴/4 + ..., one term at a time, no fancy tricks required.
The reason this is legal is that inside its interval of convergence, a power series behaves like an infinitely long polynomial. You don't need u-substitution or integration by parts on the whole thing. You apply the power rule to each piece. The result is a new power series with the same radius of convergence as the original (though the behavior at the endpoints can change, so those get checked separately when an interval of convergence is asked for). And just like any indefinite integral, you pick up a constant of integration, which you usually pin down by plugging in the center of the series.
This lives in Unit 10 (Infinite Sequences and Series) on the BC exam, specifically in the topics on representing functions as Taylor, Maclaurin, and power series. The CED expects you to construct new series from known ones, and term-by-term integration (along with term-by-term differentiation and substitution) is one of the sanctioned ways to do that. It's how the classic results get built. The series for ln(1+x) comes from integrating the geometric series for 1/(1+x), and the series for arctan(x) comes from integrating the series for 1/(1+x²). Memorizing every Maclaurin series would be brutal. Knowing how to integrate a known series term by term means you can derive what you need on the spot, which is exactly the skill the BC exam rewards.
Keep studying AP Calculus Unit F2YYi4J0XieQzPcd
Term-by-term Differentiation (Unit 10)
This is the mirror-image move. Differentiating a power series term by term takes you from a series to its derivative's series, while integration goes the other direction. Together they let you travel up and down the antiderivative chain starting from one known series.
Constant of Integration (Unit 6)
Every term-by-term integration of an indefinite integral produces a + C, and forgetting it is the classic point-loser. You usually find C by plugging the center of the series (often x = 0) into both sides.
Definite Integral (Unit 6)
Term-by-term integration also works between bounds. Integrating a series from 0 to x is how you generate alternating series approximations for integrals that have no elementary antiderivative, like the integral of e^(−x²).
U-Substitution (Unit 6)
Substitution is for integrating a whole closed-form function in one shot. Term-by-term integration is for when the function is already broken into a series, and the power rule on each term replaces any clever substitution.
This is BC-only material, so AB students can skip it. On the BC exam it shows up in two main ways. Multiple-choice stems hand you a power series and ask for the series of its antiderivative, or ask which series represents something like ln(1+x) or arctan(x). FRQs in Unit 10 frequently start you with a known Maclaurin series and ask you to build a new one, and integrating term by term is often the intended step. The graded skills are mechanical but unforgiving. Apply the power rule to each term correctly, adjust the general term's exponent and denominator, include the constant of integration, and state the interval of convergence (remembering that endpoints must be retested, since integration can change endpoint behavior).
They're inverse operations on the same idea, so it's easy to grab the wrong one under time pressure. Differentiation term by term multiplies each coefficient by the exponent and drops the exponent by one, making terms shrink in degree. Integration term by term divides by the new exponent and raises the degree, and it adds a constant of integration that differentiation never produces. Quick check, if the target function is an antiderivative of your known series (like ln(1+x) from 1/(1+x)), you integrate. If it's a derivative, you differentiate.
Term-by-term integration means integrating each term of a power series with the basic power rule, just like a polynomial.
It works inside the interval of convergence, and the new series keeps the same radius of convergence, though endpoint behavior can change.
Always add a constant of integration for indefinite integrals, then solve for it by plugging in the center of the series.
The Maclaurin series for ln(1+x) and arctan(x) come from term-by-term integration of geometric series, so you can derive them instead of memorizing.
This is BC-only content from Unit 10 and pairs with term-by-term differentiation as the two main ways to build new series from known ones.
It's the technique of integrating a power series one term at a time using the power rule, producing a new power series for the antiderivative. It's a core skill in Unit 10 for building Taylor and Maclaurin series from known ones.
Yes, as long as you stay inside the power series' interval of convergence. There the series acts like an infinitely long polynomial, so integrating each term is valid and the radius of convergence stays the same.
Integration raises each exponent and divides by the new power, and it adds a constant of integration. Differentiation lowers each exponent and multiplies by the old power, with no constant. They're inverse operations, so check whether your target is an antiderivative or a derivative of the series you know.
The radius of convergence stays the same, but the endpoints can flip. For example, integrating the geometric series for 1/(1+x) gives the series for ln(1+x), which converges at x = 1 even though the original doesn't. Always retest endpoints.
No. Power series are Unit 10 content, which is BC-only. If you're in AB, you won't see this. BC students should expect it in both multiple choice and series FRQs.