#7: INTEGRATION BY PARTS

1. HOMEWORK #1: In this section you will translate the product rule for

derivatives into a rule for integration called integration by parts.

a) Write the product rule for derivatives: d (uv)/dx =

b) Multiply the product rule by dx to get the product rule for differentials: d (uv) =

c) Solve for u dv =

d) Integrate both sides. ( Remember d(uv) = uv + C. )

The above rule is called the integration by parts formula. Usually the constant of integration, C, is not written. It is added in when the final results are obtained. The trick to integration by parts is figuring out which part of the integrand to make u and which part to make dv. (The dx is always part of dv.) The following problems illustrate some ways this method is used.

2. Eliminate the x. In the following let u = x.

a) x sinx dx

i) What is du, dv and v?        ii) Evaluate the integral by parts.

iii) Add the arbitrary constant.

b) x e 2x dx

i) What is du, dv and v?        ii) Evaluate the integral by parts.

iii) Add the arbitrary constant.

3. Eliminate a polynomial by repeated use of integration by parts. In the following let u be the polynomial.

a) x² sinx dx

i) (First application of parts.) Let u be the polynomial in the integral you can't integrate. What is u, du, dv and v?

ii) Evaluate the integral by parts. Simplify what you can.

iii) (Second application of parts.) Again let u be the polynomial in the integral you can't integrate. What is u, du, dv and v?

iv) Evaluate the using parts again. Simplify.        v) Add the arbitrary constant.

b) HOMEWORK #2: x³ e ^{- x} dx. Use methods similar to the above problem to work this. You will have to use integration by parts three times.

4. Eliminate lnx. Let u = lnx.

a) lnx dx

i) What is du, dv and v?        ii) Evaluate the integral by parts.

iii) Add the arbitrary constant.

b) x² lnx dx

i) What is du, dv and v?        ii) Evaluate the integral by parts.

iii) Add the arbitrary constant.

5. Eliminate inverse trig functions. Let u be the inverse trig function.

a) x tan ^-1 x dx

i) What is du, dv and v?        ii) Evaluate the integral by parts.

iii) Add the arbitrary constant.

b) HOMEWORK #3: sin ^-1 x dx

i) What is du, dv and v?        ii) Evaluate the integral by parts.

iii) Add the arbitrary constant.

6. Other forms. In these you want to get rid of the polynomial but you must choose a dv that can be integrated.

a) . Let

i) What is du, dv and v?        ii) Evaluate the integral by parts.

iii) Add the arbitrary constant.

b) HOMEWORK #4: x³ e ^{x 2} dx. Let dv = x e ^{x 2} dx

i) What is du, dv and v?        ii) Evaluate the integral by parts.

iii) Add the arbitrary constant.

7. Repeating forms. There are some functions, like sinx, cosx, e ^x, sinhx, that repeat themselves when they are integrated or differentiated. This means that if they are part of an integrand, you can sometimes repeat the original integral (with a different coefficient and no other unknown integrals) using integrations by parts and therefore solve for it algebraically. Do the following examples.

a) e ^{2 x} sinx dx

i) Let u = e ^{2 x}. What is du, dv and v?        ii) Evaluate the integral by parts.

iii) Evaluate the integral by parts.

iv) Let u = e ^{2 x} again in the new integral. What is du, dv and v?

v) Evaluate the integral by parts.

vi) Notice you now have two e ^{2 x} sinx dx integrals in your equation. Bring them both to one side of the equation and solve for it.

vii) Add the arbitrary constant.

b) HOMEWORK #5: Repeat the above method for e^3x cosx dx

HOMEWORK ASSIGNMENT:

REQUIRED PROBLEMS	#1, 2, 3, 4, 5; Section 7.2: 14, 30, 56
SUGGESTED PROBLEMS	Section 7.2: Odd 3/7, 9, 15, 23, 27, 29, 33, 43, 45, 55, 63, 75, 83

Informative web sites

• http://archives.math.utk.edu/visual.calculus/4/int_by_parts.3/ Integration by Parts by Larry Husch at University of Tennessee.
• http://archives.math.utk.edu/visual.calculus/4/int_by_parts.2/ Derivation, Integration by Parts: by Larry Husch at University of Tennessee.
• http://ccwf.cc.utexas.edu/~egumtow/calculus/II/parts.html Integration by Parts by Eric Gumtow, University of Texas
• http://chaos.math.temple.edu/cgi-bin/manager Calculus on the Web by Gerardo Mendoza and Dan Reich, Temple University. Go to book II, methods of integration, integration by parts

READING ASSIGNMENT BEFORE NEXT WORKSHEET: Section 7.3.