Integration by Parts

1. Integration by Parts If u and v are functions of x and have continuous derivatives, then

2. Steps for Integration By Parts IBP 1. Express the Integrand as a product of two functions. 2. Find u using the LIPET rule, and name the other functions times dx, dv. 3. Find du by differentiation and v by integration. 4. Substitute u, du, v, & dv into the IBP formula and evaluate the integral of vdu.

3. LIPET is your new order of selection of u for IBP L Logarithm I Inverse Trig Functions P Polynomials E Exponential Functions T Trig Functions

4. VIDEO TUTORIALS (Allow few second to load)

5. Summary of Common Integrals Using Integration by Parts 1. For integrals of the form Let u = xn and let dv = eax dx, sin ax dx, cos ax dx 2. For integrals of the form Let u = lnx, arcsin ax, or arctan ax and let dv = xn dx 3. For integrals of the form or Let u = sin bx or cos bx and let dv = eax dx

6. Evaluate To apply integration by parts, we want to write the integral in the form There are several ways to do this. u dv u dv u dv u dv Following our guidelines, we choose the first option because the derivative of u = x is the simplest and dv = ex dx is the most complicated.

7. u = x v = ex du = dx dv = ex dx u dv

8. INTERATCTIVE INTEGRATION BY PARTS WORKSHEET (Soln Next Page)

9. INTERACTIVE INTEGRATION BY PARTS WORKSHEET Allow few minutes to load. Click on problem for detailed solutions

10. Since x2 integrates easier than ln x, let u = ln x and dv = x2 u = ln x dv = x2 dx

11. Repeated application of integration by parts u = x2 v = -cos x du = 2x dx dv = sin x dx Apply integration by parts again. u = 2x du = 2 dx dv = cos x dx v = sin x

12. Repeated application of integration by parts Neither of these choices for u differentiate down to nothing, so we can let u = ex or sin x. Let’s let u = sin x v = ex du = cos x dx dv = ex dx u = cos x v = ex du = -sinx dx dv = ex dx

14. Guidelines for Integration by Parts • Try letting dv be the most complicated portion of the integrand that fits a basic integration formula. Then u will be the remaining factor(s) of the integrand. • Try letting u be the portion of the integrand whose derivative is a simpler function than u. Then dv will be the remaining factor(s) of the integrand.