# Ejercicio 9.4

## Apartado 1)

kill(all)$

v(x):=exp(x);
f(t):=log(1+t)/(1 + 3*t^3);
F1(x):=integrate(f(t), t, 0, v(x));

diff(F1(x),x);

DF1(x):=(%e^x*log(%e^x+1))/(3*%e^(3*x)+1);

u(x):=exp(-x);
f(t):=log(1+t)/(1 + 3*t^3);
F2(x):=integrate(f(t), t, 0, u(x));

diff(F2(x),x);

DF2(x):=-(%e^(-x)*log(%e^(-x)+1))/(3*%e^(-3*x)+1);

DF(x)=DF1(x) - DF2(x);

DF(x):=(%e^x*log(%e^x+1))/(3*%e^(3*x)+1)+(%e^(-x)*log(%e^(-x)+1))/(3*%e^(-3*x)+1);

wxplot2d([DF(x)], [x,-3, 3], [y, 0, 0.5], [ylabel, ""] , [legend, "DF(x)"] , [style, [lines,2]] ,
[gnuplot_preamble, "set grid;"])$

## Apartado 2)

kill(all)$

g(x):=(%e^x*log(%e^x+1))/(3*%e^(3*x)+1)+(%e^(-x)*log(%e^(-x)+1))/(3*%e^((-3)*x)+1);

g(-x);

g(-x) - g(x);

## Apartado 3)

h=h:0.2$        makelist( g(h*x), x, 0, 5);

I1 = I1 : h*( (g(0) + g(h*5))/2 + sum(g(h*x), x, 1, 4)), numer;

Integral = 2*I1;

h=h:0.1$

I2 = I2 : h*( (g(0) + g(h*10))/2 + sum(g(h*x), x, 1, 9)), numer;

Integral = 2*I2;