# 5.5 - 5.5 The Substitution Rule(Dated ex d x Hint ex 1...

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5.5: The Substitution Rule (Dated: September 25, 2011) Theorem 1 (The Substitution Rule for Indeﬁnite Integrals) If u = g ( x ) is a differentiable function whose range is an interval I and f is continuous on I, then Z f ( g ( x )) g 0 ( x ) dx = Z f ( u ) du . Proof. Let F be an antiderivative of f ; that is, F 0 = f . Since ( F g ) 0 ( x ) = F 0 ( g ( x )) g 0 ( x ) by the chain rule, it follows from FTC that Z f ( g ( x )) g 0 ( x ) dx = Z F 0 ( g ( x )) g 0 ( x ) dx = Z ( F g ) 0 ( x ) dx = F ( g ( x )) + C = F ( u ) + C = Z F 0 ( u ) du = Z f ( u ) du . Example 1 (Selected exercises from exercises 5.5.7-5.5.46) Evaluate the indeﬁnite integrals. 8*: R x 2 ( x 3 + 5) 9 dx 11: R ( x + 1) p 2 x + x 2 dx Hint: u = 2 x + x 2 12*: R x ( x 2 + 1) 2 dx Hint: u = x 2 + 1 13*: R dx 5 - 3 x dx Hint: u = 5 - 3 x 18: R sec2 θ tan2 θ d θ Hint: u = 2 θ 19*: R (ln x ) 2 x dx Hint: u = ln x 21*: R cos p t p t dt Hint: u = p t 23*: R cos θ sin 6 θ d θ Hint: u = sin θ 25: R e x p 1 + e x dx Hint: u = 1 + e

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5.5 - 5.5 The Substitution Rule(Dated ex d x Hint ex 1...

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