Integral calculation. Indefinite integral

Differentiation Rules If f(x) = x 6 +4x 2 – 18x + 90 f’(x) = 6x 5 + 8x – 18 *multiply by the power, than subtract one from the power.

Chain rule in transcendental Take for example y=e sin(x) ….let sin(x) = u gives; du/dx = cos(x) y=e u ….dy/du =e u Use chain rule formula dy/dx= e u.cos(x) = cos(x).e sin(x) Thus, assign the function that is inside of another function “u”, in this case sin(x) in inside the exponential.

Integration Anti-differentiation is known as integration The general indefinite formula is shown below,

Integration FORMULAS FOR INTEGRATION GENERAL Formulae Trigonometric Formulae Exponential and Logarithmic Formulae Linear bracket Formula

Indefinite integrals Examples ∫ x 5 + 3x 2 dx = x 6 /6 + x 3 + c ∫ 2sin (x/3) dx = 2 ∫ sin(x/3) dx = -2 x 3cos(x/3) + c ∫ x -2 dx = -x -1 + c ∫ e 2x dx = ½ e 2x + c ∫ 20 dx = 20x + c

Definite integrals 1 3 x y y = x 2 – 2x + 5 Area under curve = A A = ∫ 1 (x 2 -2x+5) dx = [x 3 /3 – x 2 + 5x] 1 = (15) – (4 1/3) = 10 2/3 units 2 3 3

Area under curves – signed area

Area Between 2 curves Area Between two curves is found by subtracting the Area of the upper curve by Area of the lower curve. This can be simplified into Area = ∫ (upper curve – lower curve) dx A = ∫ x 2 -(x 2 -25) dx OR A = 2 ∫ 0 25-x 2 -(x 2 -25) dx OR A = 4 ∫ 0 25-x 2 dx A = 83 1/3 units 2 y = x y = 25 - x

Area Between 2 curves continued… If 2 curves pass through eachother multiple times than you must split up the integrands. y1y1 y2y2 C D A = ∫ C (y 1 -y 2 )dx + ∫ 0 (y 1 -y 2 )dx D 0 A1A1 A2A2 Let A be total bounded by the curves y 1 and y 2 area, thus; A = A 1 +A 2

Integration – Area Approximation The area under a curve can be estimated by dividing the area into rectangles. Two types of which is the Left endpoint and right endpoint approximations. The average of the left and right end point methods gives the trapezoidal estimate. y y = x 2 – 2x + 5 x x LEFT RIGHT