Integrating Using Substitution
AP Calculus BC · Flashcards · Recall
AP CALCULUS BC · FLASHCARDS
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QUESTION
What is the fastest test to confirm an integrand matches
f
(
g
(
x
)
)
g
′
(
x
)
f(g(x))g'(x)
f
(
g
(
x
))
g
′
(
x
)
for substitution?
QUESTION
What is the substitution rule for indefinite integrals (the
u
u
u
-substitution formula)?
QUESTION
What is the substitution rule for definite integrals, including how to change limits?
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ANSWER
∫
a
b
f
(
g
(
x
)
)
g
′
(
x
)
d
x
=
∫
u
(
a
)
u
(
b
)
f
(
u
)
d
u
\int_a^b f(g(x))g'(x)\,dx=\int_{u(a)}^{u(b)} f(u)\,du
∫
a
b
f
(
g
(
x
))
g
′
(
x
)
d
x
=
∫
u
(
a
)
u
(
b
)
f
(
u
)
d
u
. Change limits: when
x
=
a
x=a
x
=
a
,
u
=
g
(
a
)
u=g(a)
u
=
g
(
a
)
; when
x
=
b
x=b
x
=
b
,
u
=
g
(
b
)
u=g(b)
u
=
g
(
b
)
.
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