Using semi-tensor product of matrices and the matrix expression of logic, formulas for calculating Boolean derivatives are obtained. Using this form, the solvability of Boolean algebraic equations and Boolean differential equations is considered. Its application to fault detection of combinational circuits is investigated. Then we define the Boolean integrals as the inverse of the Boolean derivative in certain sense. Two kinds of integrals are proposed. The inverse of a partial derivative with respect to x i is called the ith primitive function. The inverse of a differential form is called the indefinite integral. A necessary and sufficient condition for the existence of the indefinite integral is proved.