I'm currently studying Continuum Mechanics, and I came across with this integration, which allows the computation of a quantity called strain energy:
$$\int_\emptyset^{\varepsilon_{ij}} \sigma_{pq}d\varepsilon_{pq}=\int_\emptyset^{\varepsilon_{ij}} C_{pqkl}\varepsilon_{kl}d\varepsilon_{pq}=\frac{1}{2}C_{ijkl}\varepsilon_{kl}\varepsilon_{ij}$$
where $C_{pqkl}$ and $\varepsilon_{kl}$ are, respectively, a constant 4th-order tensor and a 2nd-order tensor.
I know how to deal with derivatives of tensors and other kind of operations but not with integrals. Can you explain me please how to obtain the second equality? Do you know any book that explains well such operations of tensors?
By the way, $C_{pqkl}$ and $\varepsilon_{kl}$ have the following symmetries:
$C_{pqkl}=C_{qpkl}=C_{pqlk}=C_{klpq}$
$\varepsilon_{kl}=\varepsilon_{lk}$