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 (3^{2}) - ln (2^{3})
= ln 9 - ln 8
Problem 2 :
Evaluate :
Solution :
Problem 3 :
Evaluate :
Solution :
Problem 4 :
Evaluate :
Solution :
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