Question

$\int_{4}^{5}\frac{1}{x(1+x)}dx=$ ________.

• A. $\log(\frac{10}{9})$
• B. $\log(5)$
• C. $\log(2)$
• D. $\log(\frac{11}{9})$
• E. $\log(\frac{13}{9})$

Hint:

$\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
This is a definite integral of a rational function. The integrand can be simplified using the method of partial fraction decomposition. After finding the indefinite integral, we will apply the Fundamental Theorem of Calculus to evaluate it over the given limits.
Step 2: Key Formula or Approach
1. Decompose the integrand \(\frac{1}{x(1+x)}\) into partial fractions: \(\frac{A}{x} + \frac{B}{1+x}\).
2. Integrate the resulting simpler fractions. The integral of \(\frac{1}{u}\) is \(\ln|u|\).
3. Evaluate the definite integral using the limits of integration.
Step 3: Detailed Explanation
1. Partial Fraction Decomposition.
Let \(\frac{1}{x(1+x)} = \frac{A}{x} + \frac{B}{1+x}\).
To find A and B, we can multiply by the common denominator \(x(1+x)\):
\[ 1 = A(1+x) + Bx \] - To find A, let \(x=0\): \(1 = A(1+0) + B(0) \implies 1 = A\).
- To find B, let \(x=-1\): \(1 = A(0) + B(-1) \implies 1 = -B \implies B = -1\).
So, the decomposition is:
\[ \frac{1}{x(1+x)} = \frac{1}{x} - \frac{1}{1+x} \] 2. Find the indefinite integral.
\[ \int \left(\frac{1}{x} - \frac{1}{1+x}\right) dx = \int \frac{1}{x} dx - \int \frac{1}{1+x} dx \] \[ = \ln|x| - \ln|1+x| + C = \ln\left|\frac{x}{1+x}\right| + C \] 3. Evaluate the definite integral.
Now we apply the limits from 3 to 5.
\[ \left[ \ln\left|\frac{x}{1+x}\right| \right]_3^5 = \ln\left(\frac{5}{1+5}\right) - \ln\left(\frac{3}{1+3}\right) \] (We can drop the absolute value since x is positive in the interval [3, 5]).
\[ = \ln\left(\frac{5}{6}\right) - \ln\left(\frac{3}{4}\right) \] Using the logarithm property \(\ln a - \ln b = \ln(a/b)\):
\[ = \ln\left(\frac{5/6}{3/4}\right) = \ln\left(\frac{5}{6} \times \frac{4}{3}\right) = \ln\left(\frac{20}{18}\right) = \ln\left(\frac{10}{9}\right) \] Step 4: Final Answer
The value of the definite integral is \(\log\left(\frac{10}{9}\right)\).