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Lesson 10 - Integration By Parts

Lesson 10 - Integration By Parts - The technique is used in...

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TOPIC TECHNIQUES OF INTEGRATION
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TECHNIQUES OF INTEGRATION 1. Integration by parts 2. Integration by trigonometric substitution 3. Integration by miscellaneous substitution 4. Integration by partial fraction
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TECHNIQUES OF INTEGRATION 1. Integration by parts
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OBJECTIVES OBJECTIVES to evaluate integrals using integration by parts
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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 Integratio n by parts formula
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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 integral must exist. The process can be used repeatedly.
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Unformatted text preview: . 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: 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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