How do you integrate #int (2x+1) /( (x2)(x^2+4))# using partial fractions?
1 Answer
Explanation:

First step in this problem is understanding that the use of partial fractions is valid since the numerator is a lesser degree of power than the denominator.

Apply Partial Fractions:
*Note that the denominator terms are Linear (left) and Quadratic (on the right), all this means is that they are terms that cannot be factorized further.
Linear form =#A/(x2)#
Quadratic form =#(Bx+C)/(x^2+4)#
Add these two terms together and set it equal to the Numerator.
Simplify by multiplying A by the denomenator of the other fraction, and the same for B:
now distribute:
*Note: The biggest thing students confuse about partial fractions is this next step, in the left hand side you should think of it as
Collect like terms:
Solve by any method, here we will use substitution
Eq1:
Eq2:
Eq3:
Substitute in Eq1 and Eq2 into Eq3 to get:
Simplify:
Solve for B:
Use B to solve for A
Use B to solve for C
Plug in A, B, and C values into original fraction:
Simplify into:
New integral is now represented by:
Remove constants to prepare for integration:
Break apart second integral (on the right) into two separate integrals (Mind the 1/8 multiplier) and remove the constants in the numerators
First integral:
Result is:
Second Integral:
Result Is:
Third Integral:
Apply inverse trig formula by identifying denominator as Arctan:
Where
Result is:
Plug these back into full equation and do not forget the
Distribute the
Simplify:
You can simplify further if you like, however this is an acceptable answer.