In many of the following problems, it will be essential to have a calculator or computer available. You may use a software package† or write a program for solving initial value problems using the improved Euler’s method algorithms on pages 127 and 128. (Remember, all trigonometric calculations are done in radians.) Since the integral y (x) with variable upper limit satisfies (for continuous f ) the initial value problem any numerical scheme that is used to approximate the solution at x = 1 will give an approximation to the definite integral Derive a formula for this approximation of the integral using(a) Euler’s method.(b) the trapezoid scheme.(c) the improved Euler’s method.

Solution Step 1:We have given the integral with variable upper limit satisfies the initial value problem Using any numerical method to approximate the solution of this at will give the approximation to the definite integralWe need to derive the formula for this approximation of the integral using (a)Euler's method,(b)The trapezoid scheme (c)improved Euler’s method