Chapter 17: Problem 3

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{1}{(x+1)(x-5)} \mathrm{d} x $$

Short Answer

Expert verified

The technique used in the solution is partial fraction decomposition. The final values of the constants are A = -1/6 and B = 1/6.

Step by step solution

Partial Fraction Decomposition

To perform partial fraction decomposition, we express the integrand as a sum of two simpler fractions. In this case, as the denominator has two distinct linear factors, the decomposition would look like: $$ \frac{1}{(x+1)(x-5)} = \frac{A}{x+1} + \frac{B}{x-5}, $$ where A and B are constants that we need to find.

Finding Constants A and B

To find the values of A and B, we will clear the denominators of the fractions by multiplying the equation with the common denominator, $(x+1)(x-5)$: $$ 1 = A(x-5) + B(x+1). $$ Now, we will solve this equation for A and B, by equating the coefficients of the powers of x: 1) Choose x=-1, then we get: $$ 1 = A(-1-5), $$ from which, we find the value of A: $$ A = -\frac{1}{6}. $$ 2) Choose x=5, then we get: $$ 1 = B(5+1), $$ from which, we find the value of B: $$ B = \frac{1}{6}. $$

Rewrite the Integral

Now that we have found the values of A and B, we can rewrite the integral as: $$ \int \frac{1}{(x+1)(x-5)} \mathrm{d} x = \int \left(-\frac{1}{6(x+1)} + \frac{1}{6(x-5)}\right) \mathrm{d} x $$

Integrate each Term

Finally, we will integrate each term individually: $$ \int \left(-\frac{1}{6(x+1)} + \frac{1}{6(x-5)}\right) \mathrm{d} x = -\frac{1}{6}\int \frac{1}{x+1} \mathrm{d} x + \frac{1}{6}\int \frac{1}{x-5} \mathrm{d} x $$ Both of the integrals on the right-hand side are natural logarithm functions. Thus, the final result is: $$ -\frac{1}{6} \ln|x+1| + \frac{1}{6} \ln|x-5| + C $$ where C is the constant of integration.

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By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{1}{(x+1)(x-5)} \mathrm{d} x $$

Short Answer

Step by step solution

Partial Fraction Decomposition

Finding Constants A and B

Rewrite the Integral

Integrate each Term

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