Lesson 11 Integration By Parts

# Lesson 11 Integration By Parts - integral must exist. The...

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Click to edit Master subtitle style TOPIC TECHNIQUES OF INTEGRATION

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Click to edit Master subtitle style TECHNIQUES OF INTEGRATION 1. Integration by parts 2. Integration by trigonometric substitution 3. Integration by miscellaneous substitution 4. Integration by partial fraction
Click to edit Master subtitle style TECHNIQUES OF INTEGRATION 1. Integration by parts

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OBJECTIVES to evaluate integrals using integration by parts
Integration by Parts : It is derived from the differentials of the product of two factors. If u and v are both differentiable functions of x, then d(uv) = udv + vdu The most useful among the techniques of integration is the integration by parts.

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d(uv) = udv + vdu By transposition, udv = d(uv) – vdu Integrating both sides of the equation, we have - = vdu uv udv Integrati on by parts formula
The integral is expressed in terms of another integral which must be simpler than the given integral, and is easier to evaluate. udv vdu Thus, given an integrand, a factor may be set as u, which is differentiable, and the other part as dv where its

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Unformatted text preview: integral must exist. The process can be used repeatedly. . The technique is used in integrating odd powers of : • odd powers secant, cosecant, hyperbolic secant and hyperbolic cosecant like , • inverses of trigonometric and hyperbolic functions like, ∫ xdx 4 sec 3 ∫ dx x h csc x 2 5 ∫-xdx 2 sin 1 ∫-xdx 3 cosh x 1 • products of transcendental /algebraic functions like ∫ xdx 4 sin x 2 ∫ xdx cos e x 2 EXAMPLE: Evaluate each of the following integrals. ∫ xdx 2 ln x . 1 ∫-xdx 2 tan x . 2 1 2 ∫-xdx 2 tan x . 3 1 2 ∫ xdx 3 cos e . 4 x 2 HOMEWORK 2-2: Evaluate each of the following integrals. ∫ θ θ θ d sin . 1 ∫ du u cos . 2 ∫ dx e x . 3 x 2 ∫--α α 1 1 1 d Cos . 4 ∫-ydy Sin . 5 1 ∫ 2 x 2 dx 3 x . 6 ∫-dz z 1 z . 7 2 3 ∫ π π-xdx 2 cos x . 8 2 ∫ ρ ρ ρ d sinh . 9 ∫ 4 1 tdt ln t . 10 ∫ dw ) w sin(ln . 11 ∫ + 1 2 x dx ) x 1 ( xe . 12 ∫-dt ) 1 t 2 ( t . 13 7 i. j. CLASSWORK ∫ xdx ln . 1 2 ∫ π π β β 4 / 3 4 / 3 d csc . 2...
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## This note was uploaded on 08/19/2011 for the course ECON 232 taught by Professor Charles during the Spring '11 term at MIT.

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Lesson 11 Integration By Parts - integral must exist. The...

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