Question

The definite integral \(\mathop \smallint \limits_1^3 \frac{1}{x}dx\) is evaluated using Trapezoidal rule with a step size of 1. The correct answer is __________

Answer 1

Concept:

By trapezoidal rule,

\(\mathop \smallint \limits_{{x_0}}^{{x_n}} f\left( x \right)dx = \frac{h}{2}\left[ {\left( {{y_0} + {y_n}} \right) + 2\left( {{y_1} + {y_2} + \ldots } \right)} \right]{{\;\;\;\;\;}} \ldots \left( 1 \right)\)

Calculation:

Given:

h = 1 &  \(y=\frac{1}{x}\)

Table for calculating y₀, y₁, y₂,  y₃ ......y_n

x	1	2	3
y = 1/x	1	\(\frac{1}{2}=0.5\)	\(\frac{1}{3}=0.3333\)
Designation	y0	y1	y2

Using equation (1),

 

\(\Rightarrow \mathop \smallint \limits_1^3 \frac{1}{x}dx = \frac{1}{2}\left[ {\left( {1 + \frac{1}{3}} \right) + 2\left( {0.5} \right)} \right] = 1.167\)