Calculus/Integration techniques/Partial Fraction Decomposition/Solutions

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Evaluate the following by the method partial fraction decomposition.

1.

Decompose the fraction:

Equate coefficients of x:

Solve the system of equations:

Rewrite the integral and solve:

2.

Decompose the fraction:

Equate coefficients:

Solve the system of equations:

Rewrite the integral and solve:

3.

Decompose the fraction:

Equate the coefficients:

Solve the system of equations:

Rewrite the integral and solve:

4.

Equate the coefficients for each power of x. For :

, for :

, and for the constant terms:

Solve the system of equations however you see fit (Gaussian elimination with back-substitution used here):

So

To evaluate the first integral use substitution, letting , .
To evaluate the second integral use substitution, letting , , .
To evaluate the third integral, use the trigonometric substitution , .

5.

Decompose the fraction:

Equate the coefficients:

Solve the system of equations:

Rewrite the integral and solve:

Making the substitution

in the second integral and

in the third integral, we have

6.

Decompose the fraction:

Equate coefficients:

Solve the system of equations any way you see fit. Here, we'll solve for by Cramer's rule, then plug in to solve for the other variables. The denominator in Cramer's rule will be

Expanding across the top row gives

Expanding across the top rows in both matrices gives

Solve the individual determinants

So

Now use Cramer's rule to solve for  :

Expanding down the first column gives

Expanding across the first row gives

Expanding down the last column gives

Now that we know , we can solve for using the first equation

We can solve for using the second equation and the value of

We can solve for using the third equation and the values we've found so far

We can solve for using the last equation and the values of and

Finally, we can check our solution using the 4th equation and the values we've found

Rewrite the integral and solve

Let's solve each integral separately. To solve the first, use the substitution

To solve the second integral, use the substitution

To solve the third integral, use the substitution

To solve the fourth integral, use the substitution

To solve the last integral, use the substitution

Putting it all together, we have