Evaluate the integral by first performing long division on the integrated and then writing the proper fraction as a sum of partial fractions.
x^4/x^2-9
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Then you can do the partial fraction and integration . . .
To evaluate the integral ∫x^4/(x^2 - 9), we can first perform long division on the numerator and denominator to simplify the fraction.
First, divide x^4 by x^2: x^4 ÷ x^2 = x^2
This gives us x^2 as the result of the division.
Next, multiply the entire denominator (x^2 - 9) by the quotient (x^2):
x^2 * (x^2 - 9) = x^4 - 9x^2
Subtract this result from the original numerator to obtain the remainder of the long division:
x^4 - (x^4 - 9x^2) = 9x^2
Now we can rewrite the integral with the long division result:
∫(x^4 - 9x^2)/(x^2 - 9) dx
Next, we need to factor the denominator x^2 - 9. This can be written as the difference of squares: (x - 3)(x + 3).
Now, we can write the proper fraction as a sum of partial fractions. Since the degree of the remainder 9x^2 is less than the degree of the denominator, the partial fraction decomposition will have linear terms in the numerator.
The partial fraction decomposition is given by:
∫9x^2/(x^2 - 9) dx = ∫[A/(x - 3) + B/(x + 3)] dx
To find the values of A and B, we can multiply both sides of the equation by (x^2 - 9) and then equate the numerators:
9x^2 = A(x + 3) + B(x - 3)
Expanding the right side:
9x^2 = (A + B)x + (3A - 3B)
Equating coefficients of like terms:
A + B = 0 (coefficient of x terms)
3A - 3B = 9 (coefficient of constant terms)
From the first equation, we can solve for A:
A = -B
Substituting this into the second equation:
3(-B) - 3B = 9
-6B = 9
B = -9/6 = -3/2
Substituting the value of B back into the first equation:
A + (-3/2) = 0
A = 3/2
Therefore, A = 3/2 and B = -3/2.
Now, we can rewrite the integral using the partial fraction decomposition:
∫(x^4 - 9x^2)/(x^2 - 9) dx = ∫[3/2(x - 3)/(x^2 - 9) - 3/2(x + 3)/(x^2 - 9)] dx
Integrating each term separately:
∫[3/2(x - 3)/(x^2 - 9)] dx = 3/2 ∫(x - 3)/(x^2 - 9) dx
∫[-3/2(x + 3)/(x^2 - 9)] dx = -3/2 ∫(x + 3)/(x^2 - 9) dx
Now we can evaluate these integrals by using a substitution or a trigonometric substitution, depending on the form of the integrals.