True or False? A product of two functions can be integrated by integrating each factor and multiplying the results.

False. The term-by-term rule works for sums and differences only, and there is no matching rule for products or quotients. Such an integrand has to be expanded, rearranged, or handled by a technique such as substitution or parts.

AP®MathsCollege BoardCalculus BCFlashcardsUnit 6: Integration & Accumulation of ChangeIntegration & Antiderivatives

Integration & Antiderivatives (College Board AP® Calculus BC): Flashcards

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FrontIndefinite Integrals
Define the indefinite integral of a function $f$ .

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Define the indefinite integral of a function $f$ .
It is written $\int f (x) d x$ , where the $\int$ symbol says integrate and the $d x$ says the integration is with respect to $x$ .
Evaluating it means finding an antiderivative of $f$ and adding a constant.
Why does an indefinite integral always need a constant of integration?
Because the derivative of a constant is zero, so if $F (x)$ is an antiderivative then so is $F (x) + C$ for every constant $C$ .
There is no unique antiderivative of a function, only a whole family of them.
Define an antiderivative of $f$ .
An antiderivative of $f$ is a function $F$ whose derivative is $f$ , so that $F^{'} (x) = f (x)$ .
Integration is the inverse of differentiation, which is why integrating $f$ returns one.
True or False?
The graphs of two antiderivatives of the same function are vertical translations of each other.
True.
Any two antiderivatives of $f$ differ only by a constant, and adding a constant shifts a graph up or down without changing its shape.
They have the same slope at every value of $x$ , namely $f (x)$ .
Fill in the two parts missing from this indefinite integral:
$\int (2 x + 1)______= x^{2} + x +______$
The completed integral is $\int (2 x + 1) d x = x^{2} + x + C$ .
Both parts are needed: the differential says which variable is being integrated, and the constant of integration is part of the answer rather than an optional extra.
Is $\int f (x) d x$ a number or a function?
A function of $x$ , not a number.
Integrating a function of $x$ produces another function of $x$ , together with an arbitrary constant.
How can you find the indefinite integral of a common function?
Reverse a derivative you already know: if $f^{'} (x) = g (x)$ then $\int g (x) d x = f (x) + C$ .
Every standard derivative result therefore has an indefinite integral counterpart.
Fill in the two missing parts of the rule for integrating a power of $x$ :
$\int x^{n} d x = \frac{1}{______} x^{______} + C$ for every $n$ except $- 1$
The completed rule is $\int x^{n} d x = \frac{1}{n + 1} x^{n + 1} + C$ for every $n$ except $- 1$ .
Raise the power by one, then divide by the new power.
Why does the power rule for integration fail when $n = - 1$ ?
Because $n + 1$ would be zero, and the rule divides by it.
That one case has to be integrated using logarithms instead.
True or False?
The indefinite integral of $\frac{1}{x}$ is $\ln x + C$ .
False.
It is $\ln | x | + C$ , with the absolute value bars.
Without them the result would only be valid for positive $x$ , whereas $\frac{1}{x}$ is defined for negative $x$ as well.
Fill in the two missing coefficients in these standard integrals:
$\int e^{k x} d x =______e^{k x} + C$ and $\int a^{k x} d x =______a^{k x} + C$ for a positive constant $a$
The completed integrals are $\int e^{k x} d x = \frac{1}{k} e^{k x} + C$ and $\int a^{k x} d x = \frac{1}{k \ln a} a^{k x} + C$ .
Each reverses a chain-rule derivative, so dividing undoes the factor that differentiating would have produced.
What are $\int \cos k x d x$ and $\int \sin k x d x$ ?
They are $\frac{1}{k} \sin k x + C$ and $- \frac{1}{k} \cos k x + C$ .
The minus sign goes with the cosine, because differentiating $\cos$ produces a minus sign that integrating has to undo.
Find $\int {(\frac{1}{\cos 3 x})}^{2} d x$ .
It is $\frac{1}{3} \tan 3 x + C$ .
Since $\sec x = \frac{1}{\cos x}$ , the integrand is really $\sec^{2} 3 x$ , and $\int \sec^{2} k x d x = \frac{1}{k} \tan k x + C$ .
What is $\int \frac{1}{k^{2} + x^{2}} d x$ ?
It is $\frac{1}{k} \arctan (\frac{x}{k}) + C$ .
With $k = 1$ this reduces to $\arctan x + C$ , which reverses the derivative of the inverse tangent.
Why does $\int \frac{1}{\sqrt{1 - x^{2}}} d x$ have two standard answers?
Because $\arcsin x$ and $- \arccos x$ both have that derivative, so both are antiderivatives of the integrand.
They differ only by a constant, and $\arcsin x + C$ is the form normally written.
How do you integrate a sum or difference of terms?
Integrate each term separately and add or subtract the results, since $\int (f (x) \pm g (x)) d x = \int f (x) d x \pm \int g (x) d x$ .
A constant multiplying a term can be taken outside its own integral and put back at the end.
Fill in the missing number in the rule for integrating a sum of several terms:
when a sum of several terms is integrated term by term, only $______$ constant of integration is needed for the whole answer
The completed rule is: when a sum of several terms is integrated term by term, only one constant of integration is needed for the whole answer.
Each term would contribute a constant of its own, but a sum of constants is just another constant.
Find $\int (3 x^{3} - 2 \sin 3 x + 5 e^{4 x}) d x$ .
It is $\frac{3}{4} x^{4} + \frac{2}{3} \cos 3 x + \frac{5}{4} e^{4 x} + C$ .
Each constant comes outside its own integral, and the middle term ends up positive because integrating $\sin$ introduces a minus sign of its own.
True or False?
A product of two functions can be integrated by integrating each factor and multiplying the results.
False.
The term-by-term rule works for sums and differences only, and there is no matching rule for products or quotients.
Such an integrand has to be expanded, rearranged, or handled by a technique such as substitution or parts.
Find $\int {(x^{2} + 2)}^{2} d x$ .
It is $\frac{1}{5} x^{5} + \frac{4}{3} x^{3} + 4 x + C$ .
Expand the bracket first, since ${(x^{2} + 2)}^{2} = x^{4} + 4 x^{2} + 4$ , and then integrate term by term.
What is the first step in finding $\int \frac{5 x^{3} - 3}{x^{2}} d x$ ?
Split the fraction over its denominator and use the laws of exponents, giving $5 x - 3 x^{- 2}$ .
Dividing each term by $x^{2}$ turns a quotient into a sum, which can then be integrated term by term.
Find $\int \frac{2 x^{3} + 5}{\sqrt{x}} d x$ .
It is $\frac{4}{7} x^{\frac{7}{2}} + 10 x^{\frac{1}{2}} + C$ .
Writing $\sqrt{x}$ as $x^{\frac{1}{2}}$ and splitting the fraction gives $2 x^{\frac{5}{2}} + 5 x^{- \frac{1}{2}}$ , and each term then integrates by the power rule.
What extra information lets you find the value of $C$ ?
The value of $F$ at one particular value of $x$ , or equivalently one point that the graph of $F$ passes through.
Either gives an equation that can be solved for $C$ .
Fill in the missing coordinate when the graph of $F$ passes through the point $(x_{0}, y_{0})$ :
$y_{0} = F (______)$ so substituting gives an equation for $C$
The completed relationship is $y_{0} = F (x_{0})$ , so substituting gives an equation for $C$ .
The $x$ -coordinate goes into the function and the $y$ -coordinate is what the resulting expression must equal.
True or False?
You should substitute the known point before integrating.
False.
Integrate first, so that a constant $C$ appears in the expression, and only then substitute the known point to find its value.
Substituting first leaves nothing for the point to determine.
$F$ satisfies $F^{'} (x) = 2 x + 3$ and its graph passes through $(1, 2)$ . Find $F (x)$ .
It is $F (x) = x^{2} + 3 x - 2$ .
Integrating gives $x^{2} + 3 x + C$ , and $F (1) = 2$ gives $1 + 3 + C = 2$ , so $C = - 2$ .
How many antiderivatives of $f$ pass through a given point?
Exactly one.
The antiderivatives of $f$ differ only by a constant, so fixing the value at a single point picks out one member of the family.
A projectile's height satisfies $h^{'} (t) = 70 - 32 t$ and $h (2) = 81$ . Find $h (t)$ .
It is $h (t) = 70 t - 16 t^{2} + 5$ .
Integrating gives $70 t - 16 t^{2} + C$ , and substituting $t = 2$ gives $140 - 64 + C = 81$ , so $C = 5$ .

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