Bonjour;
1)
a)
[tex]\int_x^{x^2}\dfrac{1}{tln(t)}dt=\int_x^{x^2}\dfrac{1}{t}\dfrac{1}{ln(t)}dt=\int_x^{x^2}(ln(t))'\dfrac{1}{ln(t)}dt\\\\\\=\int_x^{x^2}\dfrac{(ln(t))'}{ln(t)}dt=\big[ln(ln(t))\big]_x^{x^2}=ln(ln(x^2))-ln(ln(x))\\\\\\=ln(2ln(x))-ln(ln(x))=ln(2)+ln(ln(x))-ln(ln(x))=ln(2)\ .[/tex]
b)
[tex]\int_x^{x^2}\dfrac{\sqrt{t}-1}{tln(t)}dt=\int_x^{x^2}(\dfrac{\sqrt{t}}{tln(t)}-\dfrac{1}{tln(t)})dt\\\\\\=\int_x^{x^2}\dfrac{\sqrt{t}}{tln(t)}dt-\int_x^{x^2}\dfrac{1}{tln(t)}dt\\\\\\=\int_x^{x^2}\dfrac{\sqrt{t}}{tln(t)}dt-ln(2) = g(x)-ln(2)\ .[/tex]
