By William McCain
This version expands its scope as a with ease prepared petroleum fluids reference ebook for the training petroleum engineer and an authoritative collage textual content.
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J c o s u du = sin u 15. 16. 1cot u du = log, sin u +c u+c + C = -log, cos u + C = -log, csc tan u du = log, sec u [ 17. sec u du = log,(sec u + tan u) + C = log, tan 18. csc u du = log,(csc u - cot = log, tan 2 19. sinz u du = 2 20. cos2 u du = 2 21. sec2 u du = tan u + 22. cscz u du = -cot u + C 23. tanz u du = tan u 24. 25. 1 1 u- +C U 51 sin u cos u + c 1 + 5 u u) sin u cos u c -u+C u-u+c j ctn2 u du = -cot U 1 = 2 tan-' - + c j u du +a a +c + t )+ +C C Differential and Integral Calculus u-a du a 27.
B2- 4ac 2a The sum of the roots is -b/a and their product is c/a. Third-degree equations (cubic equations), in the general case, have the form, after division by the coefficient of the highest-order term, xJ + ax2 + bx + c = 0 with the solution x: = Ax, + B where x, A B = = = x - a/3 3(a/3)2 - b -2(a/3)3 + b(a/3) - c Exponential equations are of the form ax = b with the solution x = (log b)/(log a) and the root (log b)/(log a). The complete logarithm must be taken, not only the mantissa. Trigonometric equations are of the form a cos x * b sin x If an acute angle u is found, where tan u = b/a and an angle v (0' < v < 180') is found, where cos2 v = c2/(az + b2) the solution is x = f ( u f v) and the roots are +(u + v) and +(u - v), depending on the sign of b.
Un form a fundamental system of solutions to the homogeneous equation [E(x) t zero]. A set of functions has linear independence if its Wronskian determinant, W(x), # 0, where UI W(x) = u, . up up . u;" u; ... u, ... U" ... ... u: and m = n - lLh derivative. ) The Laplace Transformation The Laplace transformation is based upon the Laplace integral which transforms a differential equation expressed in terms of time to an equation expressed in terms of a complex variable B + j w . The new equation may be manipulated algebraically to solve for the desired quantity as an explicit function of the complex variable.