Integration by Parts - Bruce E. Shapiro

More documents

Recommendations

Info

$NAME: Math 103L: Exercise on Functions ... - Bruce E. Shapiro$

$Introducing Latex - Bruce E. Shapiro$

$Applied Differential Equations, Math 280 at CSUN - Bruce E. Shapiro$

$Math 103 Section 1.2: Linear Equations and ... - Bruce E. Shapiro$

Math 150A TOPIC 6. INTEGRATION BY PARTS ∫ Example 6.1 x sin x dx Let u = x =⇒ dv = dx (6.7a) ∫ dv = sin x =⇒ v = sin xdx = − cos x (6.7b) Then ∫ ∫ x sin x dx = uv − v du ∫ = (x)(− cos x) − (6.8a) (− cos x)dx (6.8b) = −x cos x + sin x + C (6.8c) ∫ An analogous formula holds for x cos x: ∫ x cos x dx = cos x + x sin x + C (6.9) ∫ Example 6.2 xe x dx Let Then u = x =⇒ du = dx (6.10a) dv = e x dx =⇒ v = e x (6.10b) ∫ ∫ xe x dx = (x)(e x ) − (e x )dx (6.11a) ∫ Example 6.3 Let ln x dx = xe x − e x + C (6.11b) Then ∫ u = ln x =⇒ du = dx x (6.12a) dv = dx =⇒ v = x (6.12b) ∫ ln x dx = x ln x − ( ) dx x = x ln x − x + C (6.13) x Page 16 « 2012. Last revised: February 26, 2013
TOPIC 6. INTEGRATION BY PARTS Math 150A When there are limits of integration we apply them at each step of the formula. Example 6.4 ∫ 2 1 x ln x dx Let u = ln x =⇒ du = dx x (6.14a) dv = xdx =⇒ v = 1 2 x2 (6.14b) Then ∫ 2 1 ( )∣ 1 ∣∣∣ 2 x ln x dx = (ln x) x x2 = 2 ln 2 − 1 2 ∫ 2 1 = ln 2 2 − 1 4 x2 ∣ ∣∣∣ 2 1 1 − xdx ∫ 2 1 ( 1 2 x2 ) ( dx x = ln 4 − 1 4 × 4 + 1 4 × 1 = ln 4 − 3 4 ) (6.15a) (6.15b) (6.15c) (6.15d) Sometimes integration by parts will return the original function to be integration, but reversed in sign, or multiplied by a constant. When this happens, one can write an algebraic equation for the result, as illustrated in the following example. ∫ Example 6.5 e x sin x dx Let u = e x =⇒ du = e x dx (6.16a) dv = sin x dx =⇒ v = − cos x (6.16b) Then ∫ uv− ∫ vdu { }} ∫ { e x sin x dx = −e x cos x + e x cos x dx (6.17) The second integral on the right doesn’t look any better than what we started with; nevertheless, we can continue, integrating by parts a second time, with: u = e x =⇒ du = e x dx (6.18a) dv = cos x dx =⇒ v = sin x (6.18b) « 2012. Last revised: February 26, 2013 Page 17
Page 1: Topic 6 Integration by Parts Integr
Page 5 and 6: TOPIC 6. INTEGRATION BY PARTS Math

Integration by Parts - Bruce E. Shapiro

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?