lec01 part 2

# lec01 part 2 - g x f x dx Must be able to integrate g x...

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1 - Integration by Parts Per-Olof Persson Department of Mathematics University of California, Berkeley Math 1B Calculus

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Integrals Indeﬁnite Integrals Z f ( x ) dx = F ( x ) means F 0 ( x ) = f ( x ) Result is a function (or a family of functions) Deﬁnite Integrals, Fundamental Theorem of Calculus Z b a f ( x ) dx = ±Z f ( x ) dx ² b a Result is a number
The Substitution Rule The Substitution Rule Z f ( g ( x )) g 0 ( x ) dx = ± u = g ( x ) du = g 0 ( x ) dx ² = Z f ( u ) du Example Z 2 xe x 2 dx = ± u = x 2 du = 2 xdx ² = Z e u du = e u + C = e x 2 + C

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Integration by Parts Integration by Parts Z f ( x ) g 0 ( x ) dx = f ( x ) g ( x ) - Z g ( x ) f 0 ( x ) dx Proof. Recall product rule for diﬀerentiation : d dx [ f ( x ) g ( x )] = f ( x ) g 0 ( x ) + g ( x ) f 0 ( x ) Integrate: f ( x ) g ( x ) = Z ± f ( x ) g 0 ( x ) + g ( x ) f 0 ( x ) ² dx Rearrange: Z f ( x ) g 0 ( x ) dx = f ( x ) g ( x ) - Z g ( x ) f 0 ( x ) dx
Integration by Parts Integration by Parts Z f ( x ) g 0 ( x ) dx = f ( x ) g ( x ) - Z g ( x ) f 0 ( x ) dx Alternative Notation Let u = f ( x ) and v = g ( x ) . Then du = f 0 ( x ) dx and dv = g 0 ( x ) dx . The integration by parts formula becomes: Z udv = uv - Z v du

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Integration by Parts Example Z xe x dx = ± f ( x ) = x, f 0 ( x ) = 1 g 0 ( x ) = e x , g ( x ) = e x ² = xe x - Z e x dx = xe x - e x + C = e x ( x - 1) + C Example Z ln xdx = ± f ( x ) = ln x g 0 ( x ) = 1 ² = x ln x - Z x · 1 x dx = x ln x - Z 1 · dx = x ln x - x + C
How to Choose f ( x ) and g 0 ( x ) Integration by Parts Z f ( x ) g 0 ( x ) dx = f ( x ) g ( x ) - Z

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Unformatted text preview: g ( x ) f ( x ) dx Must be able to integrate g ( x ) Must be able to integrate g ( x ) f ( x ) LIATE rule Choose f ( x ) as high up as possible: Logarithmic (e.g. ln x ) Inverse Trigonometric (e.g. sin-1 x ) Algebraic (e.g. x-3 ) Trigonometric (e.g. sin x ) Exponential (e.g. e x ) Deﬁnite Integrals Deﬁnite Integrals Z b a f ( x ) g ( x ) dx = [ f ( x ) g ( x )] b a-Z b a g ( x ) f ( x ) dx Example Z 9 4 ln y √ y dy = " f ( y ) = ln y, f ( y ) = 1 y g ( y ) = 1 √ y , g ( y ) = 2 √ y # = [ln y · 2 √ y ] 9 4-Z 9 4 2 √ y 1 y dy = 6 ln 9-4 ln 4-Z 9 4 2 √ y dy = 6 ln 9-4 ln 4-[4 √ y ] 9 4 = 6 ln 9-4 ln 4-4...
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lec01 part 2 - g x f x dx Must be able to integrate g x...

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