Finding the nth number of an Arithmetic Sequence

INTEGRATION

➪ Integration Key Formula ;

Definition Integration written F(x), is the reversed process of differentiation f '(x) (also derivatives). Functions and their Nature Integration F(x) Function f(x) Differentiation f '(x) 1 4 · x4 ↢↢↢ x3 ↣↣↣ 3x² Constant of Integration Let's consider three functions, defined as follow: f(x) = x5,          its derivative is: f '(x) = 5x4 f(x) = x5 + 2,    its derivative is: f '(x) = 5x4 f(x) = x5 − 12,  its derivative is: f '(x) = 5x4 As you can notice, the derivatives of the three functions are the same and the constants hanging behing the two last functions have disappeared or equal to Zero (0)!!!     ஃ    we can conclude that, the derivative of any constant number equals to 0    ⇒   f(x) = C (also K) and f '(x) = 0 i.e.     i)    f(x) = 7 and f '(x) = 0     ii)    f(x) = −10 and f '(x) = 0     iii)    f(x) = − 1 2 and f '(x) = 0     iv)    f(x) = 11 3 and f '(x) = 0 As we've seen above, the three functions have the same derivative. Now if we go integrating the three of them, we actually need to precise they all have a constant to be sure to find the original functions. The general formula is: ∫ f '(x) dx = f(x) + C So for our three functions we'll have: ∫ 5x4 dx = x5 + C, with C in this case equals to 0 and the function f(x) = x5 ∫ 5x4 dx = x5 + C, with C in this case equals to 2 and the function f(x) = x5 + 2 ∫ 5x4 dx = x5 + C, with C in this case equals to −12 and the function f(x) = x5 − 12 How to find the constant of integration? Integrating a sum or a difference of functions It's about integrating by term or separetly. i.e.    1. Find ∫ x² − x √x dx    ∫ x² − x √x dx = ∫ ( x² x1/2 − x x1/2 ) dx = ∫ x3/2 dx − ∫ x1/2 dx                      = 2 5 x5/2 − 2 3 x3/2 + C    2. Find the integral of 1 + x7 + 1 x² − √x    ∫ ( 1 + x7 + 1 x² − √x ) dx = ∫ ( 1 + x7 + x−2 −x1/2 )                                            = ∫ 1 dx + ∫ x7 dx + ∫ x− 2 dx + ∫ x1/2 dx                                            = x + 1 8 x8 − 1 x − 2 3 x3/2 + C Integrating terms to the power of n: (ax + b)n i.e. Integrate f(x) = (2x + 3)4 to integrate this function, we'll proceed by substitution Let u = 2x + 3   ⇒   u4 = (2x + 3)4, then, we derivate u prior to x: du = 2 dx 1 2 du = dx then after, we derivate u4: 1 2 ∫ u4 du = 1 2 · 1 5 u5 now we replace u by its substituted value: u = 2x + 3: 1 2·5 (2x + 3)5 So,     ∫ (2x + 3)4 dx = 1 2·5 (2x + 3)5 Finding the constant of integration Assume we have a function f(x) = x3 + C, its derivative is f '(x) = dy/dx = 3x2. To find C, we need to know a point on the curve of the function. Let's say P(1, 3) that point on the curve with an equation dy/dx = 3x2 + 4                dy dx = 3x2 + 4         ⇒     y = ∫ (3x2 + 4) dx         ∴     y = x3 + 4x + C         ∴     as (1, 3) is on the curve, we can input x=1 and y=3 in x3 + 4x + C,                to find C:   3 = 1 + 4 + C     ⇒         C = − 2         ∴     y = x3 + 4x − 2 Key Formula