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5.5 Indefinite integrals and sub rule

# 5.5 Indefinite integrals and sub rule - 1 5.5 Indeﬁnite...

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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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5.5 Indefinite integrals and sub rule - 1 5.5 Indeﬁnite...

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