GATE (TF) Textile 2022 Question Paper Solution

Assuming the step size h = 1, the numeric value (rounded off to 2 decimal places) of the definite integral

$\int_{1}^{3} \frac{x}{1+x} dx$

obtained using Simpson’s rule is _____.

[Show Answer]

Ans: -1.29 to 1.33
Detailed Solution :
Given $\int_{1}^{3} \frac{x}{1+x} dx$

Step size : h =1

Simpson’s rule
$\Rightarrow \frac{h}{3} \left [ \left ( y_0 + y_n \right ) + 2\left ( y_e \right ) + 4\left ( y_0 \right )\right ]$

$y = \frac{x}{1+x}$

$= \frac{1}{3}\left [ (y_0 + y_3) + 2 (y_2) + 4(y_1)\right ]$

$= \frac{1}{3}\left [ (0 + \frac{3}{4}) + 2 (\frac{2}{3}) + 4(\frac{1}{2})\right ]$

$= \frac{1}{3}\left [ \frac{3}{4} + \frac{4}{3} + \frac{4}{2}\right ]$

$= \frac{1}{3} \left [ \frac{9+16+24}{12} \right ]$

$= 1.30$