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Integration by partial fractions

6. Solve for :
10. Evaluate:

11. Evaluate:

19. Express the improper rational function as the sum of a polynomial and a proper rational function:

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Express the improper rational function as the sum of a polynomial and a proper rational function.

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Integration by partial fractions

6. Solve for :
To solve this problem, we multiply both sides by dx and then integrate to get

or
.
Now we use partial fraction decomposition to separate the rational function into two more simple rational functions. Since both terms in the denominator are linear terms, we will set up the decomposition in this way:
.
Multiplying both sides of this equation by the factors and , we get

.
Condensing this equation to the first and last parts gives us

and we now set the coefficients of the x's from both sides equal to each other and the constant terms equal to each other to get the equations and . Solving the first of these two equations for A gives and substituting this into the second equation gives which has a ...

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