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**Unformatted text preview: **1 5.5 Indeﬁnite Integrals and Substitution Rule
Fundamental Theorem of Calculus shows that antiderivatives are the keys
to evaluation of deﬁnite integrals. In this section we study a technique
of ﬁnding antiderivatives for functions that are not elementary, but have a
special form.
Example We begin by doing things backward: diﬀerentiate sin(x2 )
sin(x2 ) =
Let’s write it again, but in slightly diﬀerent way:
sin(x2 ) =
Integrating (or anti-diﬀerentiating), we get:
sin(x2 ) dx = Thus
u=x2 cos(x2 ) · 2x dx =
Theorem Substitution Rule (u-Substitution)
If u = g (x) is a
and f is a function that is function whose range is an interval I ,
on I, then f g (x) · g (x)dx =
where F is an antiderivative of f .
Note: u-substitution is noting more than an inverse of the Chain Rule.
Proof:
d
F (g (x)) =
dx
Integrating, we get:
f (g (x))·g (x) dx = 2 Steps for u-substitution:
• Choose your u. Separate the integrand into the parts that can be
expressed in terms of u and a part that is a constant multiple of u
• Make substitution and carry out integration
• Substitute back for u to obtain the expression in terms of the original
variable
Example 1 Integrate:
• 1
dx =
ex • sin(7t) dt = • tan(s) ds = • ex tan(ex ) ln (cos(ex )) dx = 3 Example 2 Integrate:
• • √
x2 x2 + 1 dx = • Example 3
To do √
x x2 + 1 dx = √
x3 x2 + 1 dx = Integrate:
• et
dt =
1 + et • tan−1 (x)
dx =
1 + x2 4 Example 3
continued Integrate:
1 + 4x
dx =
1 + 2 x + 4 x2 • √ • cos2 (2y + 3) sin(2y + 3) dy = • cot(z ) dz = In some cases, simpliﬁcation needs to be done prior to application of u-substitution.
Example 4
• 1+x
dx =
1 + x2 • √ • √
x x + 1 dx = x2
dx =
1−x ...

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