Integral Calculus Review

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Questions and Answers

What is the result of an indefinite integral?

  • A single numerical value.
  • The derivative of the function.
  • A function.
  • A function plus a constant of integration. (correct)

The power rule for integration states that $∫x^n dx = (x^{n+1})/(n+1) + C$ is valid for all real numbers n.

False (B)

What is the integral of $1/x$ with respect to $x$?

ln|x| + C

The formula for integration by parts is ∫u dv = ______ - ∫v du

<p>uv</p> Signup and view all the answers

Match the trigonometric integral with the appropriate substitution technique:

<p>∫sin^m(x)cos^n(x) dx, where m is odd = Save a sin(x) factor and use $sin^2(x) = 1 - cos^2(x)$ ∫sin^m(x)cos^n(x) dx, where n is odd = Save a cos(x) factor and use $cos^2(x) = 1 - sin^2(x)$ ∫sin^m(x)cos^n(x) dx, where both m and n are even = Use half-angle identities</p> Signup and view all the answers

For an integral containing the expression $√(a^2 - x^2)$, which trigonometric substitution is most appropriate?

<p>$x = a sin(θ)$ (A)</p> Signup and view all the answers

When using partial fractions to integrate a rational function with a denominator that has repeated linear factors, such as P(x)/(x-a)^2, the correct decomposition is P(x)/(x-a)^2 = A/(x-a)

<p>False (B)</p> Signup and view all the answers

State Part 2 of the Fundamental Theorem of Calculus.

<p>∫[a to b] f(x) dx = F(b) - F(a), where F(x) is an antiderivative of f(x)</p> Signup and view all the answers

An improper integral with an infinite limit of integration is evaluated by taking the ______ as the limit approaches infinity.

<p>integral</p> Signup and view all the answers

Which method is used to find the volume of a solid of revolution by summing the volumes of thin disks?

<p>Disk Method (B)</p> Signup and view all the answers

Flashcards

Indefinite Integral

The set of all antiderivatives of a function.

Power Rule for Integration

∫x^n dx = (x^(n+1))/(n+1) + C, for n ≠ -1

Constant Multiple Rule

∫cf(x) dx = c∫f(x) dx; a constant can be moved outside the integral.

Sum/Difference Rule

∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx; Integrate term by term.

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u-Substitution

A technique for simplifying integrals by changing variables.

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Definite Integral

Represents the net signed area under a curve between limits.

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Fundamental Theorem of Calculus (Part 2)

∫[a to b] f(x) dx = F(b) - F(a), where F'(x) = f(x)

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Integration by Parts

Used to integrate products of functions: ∫u dv = uv - ∫v du

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Trigonometric Substitution

A method using trig functions to simplify integrals with square roots.

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Partial Fractions

Used to integrate rational functions by decomposing them into simpler fractions.

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