Is it it
$\int_0^t \left ( \int_v^t S(u)du \right ) \alpha(v)\lambda(v)dv$ =
$\int_0^t \frac{\left ( { S(t^2)- S(v^2) }\right )}{2}\alpha(v)\lambda(v)dv$
$\int_0^t \frac{\left ( { S(t^2)}\right )}{2} \alpha(v)\lambda(v)dv$ - $\int_0^t \frac{\left ( {S(v^2) }\right )}{2} \alpha(v)\lambda(v)dv$
$\frac{\left ( { S(t^2)}\right )}{2}\int_0^t \alpha(v)\lambda(v)dv$ - $\int_0^t \frac{\left ( { S(v^2) }\right )}{2} \alpha(v)\lambda(v)dv$
or
$\int_0^t \frac{\left ( { S(t)- S(v^2) }\right )}{2}\alpha(v)\lambda(v)dv$
$\int_0^t \frac{\left ( { S(t)}\right )^2}{2} \alpha(v)\lambda(v)dv$ - $\int_0^t \frac{\left ( {S(v) }\right )^2}{2} \alpha(v)\lambda(v)dv$
$\frac{\left ( { S(t)}\right )^2}{2}\int_0^t \alpha(v)\lambda(v)dv$ - $\int_0^t \frac{\left ( { S(v) }\right )^2}{2} \alpha(v)\lambda(v)dv$
Please let me know if i am doing anything wrong or missing any steps.