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Chapter 12: Problem 16

Use partial fractions to find the indefinite integral. $$ \int \frac{-4}{x^{2}-4} d x $$

Short Answer

Expert verified
\[-\ln |x-2| + \ln |x + 2 | + C\]

Step by step solution

01

Decomposition into Partial Fractions

Rewrite the rational function as the sum of its simpler fractions. To do this, it's necessary to factorize the denominator. \(x^{2}-4\) can be factorized into \((x-2)(x+2)\). Hence, the integrand can be rewritten using partial fraction decomposition as: \[-\frac{4}{x^{2}-4} = \frac{A}{x-2} + \frac{B}{x+2}\]. Multiplying through by the common denominator gives \(-4 = A(x+2) + B(x-2)\]. Let's solve for A and B by comparing coefficients.
02

Solve for A and B

Setting \(x=2\) in the equation \(-4 = A(x+2) + B(x-2)\) gives \(A(4) = -4\), hence \(A=-1\). Setting \(x=-2\) gives \(B(-4) = -4\), hence \(B=1\). Hence now, the integral becomes: \[ \int \frac{-1}{x-2} + \frac{1}{x+2} dx \].
03

Integrate the Simplified Fraction

Now, the integration can be performed on each simpler fraction separately: \[= - \int \frac{1}{x-2} dx + \int \frac{1}{x+2} dx\] which are in the form of \(\int \frac{1}{x-a} dx\), whose integral is \(\ln |x-a|\), where \(\ln\) is the natural logarithm. Therefore, the final answer is: \[-\ln |x-2| + \ln |x + 2 | + C\]

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Most popular questions from this chapter

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} \frac{1}{x} d x $$

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