Solving Complex Integrals Using Integration by Partial Fractions

Integration by partial fractions is a powerful technique used to solve complex integrals by decomposing the integrand into simpler rational functions. This method is widely used in calculus and is particularly useful when dealing with rational functions with polynomial expressions in the numerator and denominator. In this article, we will walk through the step-by-step process of solving the integral of ∫x^6 / (x^2 - 4) dx using partial fractions.

Understanding the Integral

The given integral is: ∫x^6 / (x^2 - 4) dx. To solve it using partial fractions, we first need to simplify the integrand by dividing the polynomial in the numerator by the polynomial in the denominator.

Step 1: Simplify the Integrand

Notice that we can divide x^6 by x^2 - 4 using polynomial long division.

Divide x^6 by x^2 to get x^4. Multiply x^4 by (x^2 - 4) to get x^6 - 4x^4. Subtract this from x^6 to get 4x^4. Now divide 4x^4 by x^2 to get 4x^2. Multiply 4x^2 by (x^2 - 4) to get 4x^4 - 16x^2. Subtract to get 16x^2. Now divide 16x^2 by x^2 to get 16. Multiply 16 by (x^2 - 4) to get 16x^2 - 64. Subtract to get 64.

Therefore, we can write:

x^6 / (x^2 - 4) x^4 4x^2 16 64 / (x^2 - 4)

Step 2: Set Up the Integral

Now, we rewrite the integral using the simplified form:

∫x^6 / (x^2 - 4) dx ∫(x^4 4x^2 16 64 / (x^2 - 4)) dx

Step 3: Integrate Each Term

We integrate each term separately:

∫x^4 dx x^5 / 5 ∫4x^2 dx 4x^3 / 3 ∫16 dx 16x ∫64 / (x^2 - 4) dx using partial fractions

Step 3.1: Perform Partial Fraction Decomposition

First, we decompose 64 / (x^2 - 4) into partial fractions. Note that x^2 - 4 can be factored as (x - 2)(x 2).

So, we write:

64 / (x^2 - 4) 64 / ((x - 2)(x 2)) A / (x - 2) B / (x 2)

Multiplying through by (x - 2)(x 2) gives:

64 A(x 2) B(x - 2)

Expanding and collecting like terms:

64 Ax 2A Bx - 2B

By equating coefficients, we get:

A B 0 2A - 2B 64 implies A - B 32

Solving this system:

A B 0 implies B -A A - (-A) 32 implies 2A 32 implies A 16 B -16

Thus, we have:

64 / (x^2 - 4) 16 / (x - 2) - 16 / (x 2)

Integrating these:

∫64 / (x^2 - 4) dx 16 ln(x - 2) - 16 ln(x 2) C

Step 4: Combine All Parts

Combining all parts, we get:

∫x^6 / (x^2 - 4) dx x^5 / 5 4x^3 / 3 16x 16 ln(x - 2) - 16 ln(x 2) C

Therefore, the final answer is:

∫x^6 / (x^2 - 4) dx x^5 / 5 4x^3 / 3 16x 16 ln((x - 2) / (x 2)) C

Conclusion

Integration by partial fractions is a valuable technique that simplifies complex integrals into more manageable parts. The process involves polynomial long division and partial fraction decomposition. This method not only helps in solving the integral but also in understanding the structure of the integrand.