AP CALCULUS BC : Integration of Rational Functions by Partial Fractions (Part - 1)

Problem 1 :

Evaluate :

Solution :

We can integrate the given rational function by writing it into partial fractions.

Find the values of A and B.

Multiply both sides by (x + 1)(x + 2).

x = A(x + 2) + B(x + 1)

If x = -1,

-1 = A(1) + B(0)

-1 = A + 0

-1 = A

If x = -2,

-2 = A(0) + B(-1)

-2 = 0 - B

-2 = -B

2 = B

Substitute A = -1 and B = 2 into (1).

Integrate (0 to 1) both sides with respect to x. 

= [-ln |1 + 1| + 2ln |1 + 2|] - [-ln |0 + 1| + 2ln |0 + 2|]

= [-ln 2 + 2ln 3] - [-ln 1 + 2ln 2]

= [-ln 2 + 2ln 3] - [0 + 2ln 2]

= -ln 2 + 2ln 3 - 2ln 2

= 2ln 3 - 3ln 2

= ln (32) - ln (23)

= ln 9 - ln 8

Video Lesson

Problem 2 :

Evaluate :

Solution :

Problem 3 :

Evaluate :

Solution :

Problem 4 :

Evaluate :

Solution :

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