1) Trapezoidal Rule

2) Simpson’s one third rule

3) Simpson’s three eight rule

Let $a = 0.2, b = 1.4$ and $n = 6 $

Let $y =\sin x – \log_ex + e^x\\ h=\dfrac {b-a}n=\dfrac {1.4-0.2}6=0.2$

1) Trapezoidal rule

$\int\limits_a^b f(x)dx=\dfrac h2[(y_0+y_6)+2(y_1+y_2+y_3+y_4+y_5)]\\ \therefore \int\limits_0^6 \sin x\log_e xe^x dx=\dfrac {0.2}2[(3.0295+4.702)+2(2.7975+ 2.8976+3.166+3.5598+4.0698)]\\ =0.1[7.7337+2\times 21.1949]\\ =4.0715$

2) Simpson’s one third rule

$\int\limits_a^b f(x)dx=\dfrac h3[(y_0+y_6)+4(y_1+y_2+y_5)+2(y_2+y_4)]\\ =\dfrac {0.2}3[(3.0295+4.702)+4(2.7975 + 3.166 +4.0698 ) + 2(2.8976+3.5598)]\\ =\dfrac {0.2}3[7.7337+4\times 10.033+2\times 6.4574]\\ =4.0521$

3) Simpson’s three eight rule

$\int\limits_a^b f(x)dx=\dfrac {3h}8[(y_0+y_6)+3(y_1+y_2+y_4+y_5)+2(y_3)]\\ =\dfrac {3\times 0.8}8[(3.0295+4.702)+3(2.7975 + 2.8976+3.5598 +4.0698 ) + 2(3.166)]\\ =\dfrac {0.6}8[7.7337+3\times 13.3247+2\times 3.166]\\ =4.0530$