## 1 Introduction  
Assume that \(y=y(x)\) is an unknown univariate real-valued function over \([a,b]\). Let:  
\[\Delta:=\{a=x_{0}<x_{1}<\cdots<x_{n}=b\} \tag{1}\]  
be the uniform partition of \([a,b]\) with step length \(h:=\frac{b-a}{n}\), and let:  
\[I_{j}:=\int_{x_{j}}^{x_{j+1}}y(x)dx\quad(j=0,1,\ldots,n-1) \tag{2}\]  
be the known integral values of \(y=y(x)\) over the subintervals. The interpolation function \(p=p(x)\) that satisfies:  
\[\int_{x_{j}}^{x_{j+1}}p(x)dx=I_{j}\quad(j=0,1,\ldots,n-1)\]  
is called integro interpolation. The problem arises in many fields, such as numerical analysis, mathematical statistics, environmental science, mechanics, electricity, climatology, oceanography, and so on. We refer to [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19] for its applied backgrounds and some recent developments. In this paper, we will mainly focus on the quintic spline methods; see [2; 14; 17; 18; 19] for the existing ones. The method in [2] was based on the quintic Hermite-Birkhoff polynomials. The method was very complicated because it mainly required solving two linear systems. Furthermore, besides the integral values (2), the method must use seven additional exact end conditions in terms of \(y(x_{0})\), \(y^{\prime}(x_{0})\), \(y^{\prime}(x_{1})\), \(y^{\prime}(x_{n-1})\), \(y^{\prime}(x_{n})\), \(y^{\prime\prime}(x_{0})\) and \(y^{\prime\prime^{\prime\prime}}(x_{n})\). Later, a new algorithm was given in [18] to simplify the construction of integro quintic spline. It mainly required solving two linear three-diagonal systems. It was kind of simpler than that of [2]. However, the algorithm needed five special and proper exact end conditions in terms of \(y(x_{0})\), \(y^{\prime}(x_{1})\), \(y^{\prime}(x_{n-1})\), \(y^{\prime\prime^{\prime\prime}}(x_{1})\) and \(y^{\prime\prime^{\prime\prime}}(x_{n-1})\). The method in [14] was based on quintic B-splines. It was also very simple because it took advantage of the good properties of quitic B-splines. However, five additional exact end conditions in terms of \(y(x_{0})\), \(y(x_{1})\), \(y(x_{2})\), \(y(x_{n-1})\) and \(y(x_{n})\) must be provided. In other words, these methods all need exact end conditions. This is an obvious drawback of them. New simple methods that are not dependent on exact end conditions are desired. In [17], we have studied an effective method that was not dependent on any exact end conditions. We first obtained \(n+1\) approximate function values at the knots and four approximate boundary derivative values from the integral values (2) and then used them to study a modified quintic spline interpolation problem. However, the method also had its own drawbacks. On the one hand, it needed \(n+5\) artificial values, which brought higher computational cost; on the other hand, the obtained quintic spline did not agree with the given integral values (2) over the subintervals. In [19], a local integro quintic spline method was given. It was also not dependent on exact end conditions and was able to produce good approximations. However, the obtained local integro quintic spline also did not agree with the given integral values (2) over the subintervals. Hence, these methods also need improvements. New attempts on this problem are still necessary. In this paper, we aim to develop a new effective method to overcome the above-mentioned drawbacks. We will first construct six artificial end conditions by using a similar technique to [17] and use them together with the integral values (2) to get a new kind of integro quintic spline; then, we will theoretically analyze and numerically examine the approximation properties of the new integro quintic spline. The new method is very effective, and it has the following advantages. * The method is free of any exact end conditions, and it only requires five artificial end conditions, which can be easily obtained by simple computations from several integral values. * The computational procedure of the method is concise and easy to implement.