Lesson 11 Integration By Parts

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

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Click to edit Master subtitle style TOPIC TECHNIQUES OF INTEGRATION
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Click to edit Master subtitle style TECHNIQUES OF INTEGRATION 1. Integration by parts
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
OBJECTIVES to evaluate integrals using integration by parts
Background image of page 4
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.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 08/19/2011 for the course ECON 232 taught by Professor Charles during the Spring '11 term at MIT.

Page1 / 11

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

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online