5 Steps to a 5 AP Calculus BC 2019

222 STEP 4. Review the Knowledge You Need to Score High

Each letter in the acronym represents a type of function:Logarithmic,Inverse trigono-

metric,Polynomial,Exponential, andTrigonometric. As you consider integrating by parts,

assign the factor that falls earlier in theLIPETlist asu, and the other asdv.

Example 1∫

xe−xdx

Step 1: Identifyu=xand dv=e−xdxsincex is aPolynomial, which comes before

Exponential inLIPET.

Step 2: Differentiatedu=dxand integratev=−e−x.

Step 3:

∫

xe−xdx=−xe−x−

∫

−e−xdx=−xe−x−e−x+C

Example 2∫

xsin 4xdx

Step 1: Identifyu=xanddv=sin 4xdxsincexis aPolynomial, which comes before

Trigonometric inLIPET.

Step 2: Differentiatedu=dxand integratev=

− 1

4

cos 4x.

Step 3:

∫

xsin 4xdx=

−x

4

cos 4x+

1

4

∫

cos 4xdx=

−x

4

cos 4x+

1

16

sin 4x+C

Integration by Partial Fractions

A rational function with a factorable denominator can be integrated by decomposing the

integrand into a sum of simpler fractions. Each linear factor of the denominator becomes

the denominator of one of the partial fractions.

Example 1

∫

dx

x^2 + 3 x− 4

Step 1: Factor the denominator:

∫

dx

x^2 + 3 x− 4

=

∫

dx

(x+4)(x−1)

Step 2: LetAandBrepresent the numerators of the partial fractions

1

(x+4)(x−1)

=

A

x+ 4

+

B

x− 1

.

Step 3: The algorithm for adding fractions tells us thatA(x−1)+B(x+4)=1, soAx+Bx= 0

and−A+ 4 B=1. Solving gives usA=− 0 .2 andB= 0 .2.

Step 4:

∫

dx

x^2 + 3 x− 4

=

∫

− 0. 2

x+ 4

dx+

∫

0. 2

x− 1

dx=− 0 .2ln(x+4)+ 0 .2ln

(x−1)+C