Integral, indefinite integration, methods

Methods of Integration All the methods of integration available for the indefinite integral can be used for definite integrals. In addition, several others are available for the latter integrals and are indicated below. 

Numerical Because of the property of definite integrals another method for obtaining their solution is available which cannot be apphed to indefinite integrals. This involves a numerical approximation based on the previously outlined summation definition ... 

Eoint e, n (recognizing that es = e0 = 1). However, this method works etter, using an indefinite integration of dt ... 

These suggest the existence of an indefinite series of exponentially decreasing terms Ek N and, in fact, our method of proving the theorem17 entails a matrix iterative procedure which would in principle yield a complete convergent expansion for In DN(c). An alternative integral expression for the leading correction term is... 

The second integration technique, known as the substitution method, derives from the inversion of the chain rule for differentiation described in Chapter 4. The objective here, once again, is to transform the integrand into a simpler or, preferably, a standard form. However, just like the integration by parts method, there is usually a choice of substitutions and although, in some cases, different substitutions yield different answers, these answers must only differ by a constant (remember that, for an indefinite integral, the answer is determined by inclusion of a constant). The substitution method is best illustrated using a worked problem. 

The study notes, A substantial risk premium may thus be applied by potential hydrogen infrastructure investors. In this analysis, it takes ten years for an investment in infrastructure to achieve a positive cash flow, far too long for the vast majority of investors, and, to achieve this result, significant technological advances will be required in reformers and electrolyzers, compressors, and overall systems integration as well as mass production methods for that equipment. Also, even a small excise tax on hydrogen (to make up for the revenue lost from gasoline taxes) appears to delay positive cash flow indefinitely.59... 

This method of writing indicates that the expression following the integration sign is to be integrated indefinitely, and the upper limit to be inserted. 

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