Lecture 2 - Given by Alan Hegarty PDF

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Given by Alan Hegarty...

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MA4602 – integration Alan Hegarty More standard integrals To integrate a constant that is raised to the power of a variable (ax) (note that a must be a positive number in this,) we take the original function (ax) and divide it by the natural log of the constant, adding C. (Int(ax) = ax / ln(a) + c.) e.g – int(7x) = 7x / ln(7) + c Inverse trig functions Inverse trig functions are essentially the opposite of trig functions. Sin(x) = y where y is the value when we insert an angle into x. An inverse trig function is used to determine what the angle was, making use of the trig sign and the end result. In layman’s terms, if Sin of an angle was 1, we use sin-1(1) to determine what the angle is. Sin-1 (sin (x)) = x. The inverse is true also. Tan (sin-1(x)) = x/ (1 – x2)0.5 Cos-1(Cos(x)) = x. The inverse is true also. Sin-1(x) + Cos-1(x) = pi/2 or 90o. Sin(Cos-1(x)) = (1-x2)0.5

Derivative of trig functions der(Sin-1(x)) = 1/(1-x2)0.5 der(Sin-1(x/a)) = 1/(a2-x2)0.5 der(Cos-1(x/a)) = -1/(a2-x2)0.5 der(Tan-1(x/a)) = a/(x2+a2)

Note that integrating what is found on the right hand side of the equals sign will give back what is on the left – meaning if you integrate 1/(1-x2)0.5 you will get Sin-1(x.)...