Dr. Omar Al Jadaan Assistant Professor – Computer Science & Mathematics General Education Department Mathematics

INTEGATION AS THE REVERSE OF DIFFERENTIATION INTEGRATION AND APPLICATIONS

The Power Rule for Integration The derivative of y = x 2 /2 + c is dy/dx = x So dy = x dx The integration ∫dy = ∫x dx So the answer:y = x 2 /2 + c In general the integral of ∫ dy = ∫ x n dx is y = x n+1 /(n+1) + c

Examples 1. Integrate the following using power rule: a) ∫5x 4 dx b) ∫ 4/x 5 dx Solution: a) n = 4 and n+1 = 5. So 5 x 5 /5 +c = x 5 + c b) n= -5 and n + 1 = -4. So 4 x -4 /-4 +c = - x -4 + c

Skill Practice Integrate the following: a) ∫ 8 x 7 dx b) ∫ 7 / x 8 dx c) ∫ (3x 2 + 2x + 3) dx

Integration of the Natural Exponential Function ∫ e x dx = e x + c Example: ∫ (7e x + 4)dx = 7e x + 4x + c Example: ∫ e 5x+3 dx. 1. let u = 5x +3 then it becomes ∫ e u dx 2. du = 5 dx. So dx = du/5. 3. ∫ e u dx = ∫ e u du/5 = e u /5 + c e (5x+3) /5 + c

∫ du/u= ln u Example ∫ 1/(3x – 4) dx Let u = 3x – 4, so that ∫ 1/u dx. du = 3 dx, so that dx = du/3 ∫ 1/u dx = ∫ 1/u du/3=ln u /3 + c = ln (3x -4) /3 +c

Skill practice 1. ∫ (4x -5 )3 dx 2. ∫ 1/(5x + 4) dx 3. ∫ e(3x-2) dx 4. ∫ 1/(3x + 9) dx

First Order Differential Equation A differential equation is an equation which contains derivative. For example, dy/dx = 10x. This equation is called first order differential equation because the highest order derivative is one. Solution of the differential equation dy/dx = f(x) Step 1: write the D.E. in the form of dy/dx = RHS Step 2: integrate both sides with respect x. Step 3: if conditions are given for x and y, substitute these values and solve for the arbitrary constant c. Step 4: find the value of c and find the particular solution.

Find the solution for the differential equation dy/dx – 6x + 2 = 0, given that x = 3 when y = 0. Step 1. dy/dx = RHS dy/dx = 6x - 2 Step 2. ∫ dy = ∫ (6x -2)dx Step 3. the solution y = 3x 2 – 2x + c Step 4: substitute the values 0 = 27 – 6 + c, c = -21 then y = 3x 2 – 2x – 21. Examples

Skill practice Find the solution for each of the following differential equations: 1. dy/dx = 5x, given x=2 when y = dy/dx = e x, given x=0, when y=2. 3. dy/dx = 1/x, given x = e when y =1.

4. dy/dx = sin x, given x = 0 when y =1. 5. dy/dx = cos x, given x = 0 when y =1.

Integration Rules In the following formulas, let a, b, c, n be constants, and General Formulas