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
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