FTC Review The Method of Substitution

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FTC ReviewThe Method of Substitution

• February 4, 2004

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The Definite Integral as Area

• Let f be a continuous function defined on the
  interval a, b. The definite integral of f from
  a to b, denoted by
• represents the total signed area of the region
  bounded by y f (x), the vertical lines x a
  and x b, and the x-axis.

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Properties of Definite Integrals
Let f and g be continuous functions defined on
the interval a, b. Furthermore, let c and k be
constants such that a lt c lt b. Then

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The Fundamental Theorem of Calculus
Let f be a continuous function defined on a,
b, and let F be any antiderivative of f. Then
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Keeping It Straight
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Substitution Rule for Indefinite Integrals
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Implementing the Substitution Rule

• Choose u.
• Differentiate u w.r.t. x and solve for du.
• Substitute u and du into the old integral
  involving x to form a new integral involving only
  u.
• Antidifferentiate with respect to u.
• Re-substitute to find the antiderivative as a
  function of x.

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Two Special Forms
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Substitution Rule for Definite Integrals
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Implementing the Substitution Rule(Definite
Integrals)

• Choose u g(x).
• Differentiate u w.r.t. x and solve for du.
• Substitute u and du into the old integral
  involving x, as well as converting endpoints from
  a and b to g(a) and g(b).
• Antidifferentiate with respect to u and evaluate
  at the new endpoints.

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Arcsine (Inverse Sine Function)

• For x in -1, 1, y arcsin x is defined by the
  conditions
• x sin y and
• ?/2 ? y ? ?/2.
• In words, arcsin x is the angle between ?/2 and
  ?/2 whose sine is x.