**HP Prime: Solving Integral Equations**

**Introduction**

The program INTEGRALSOLVE solves the following integral equation for x:

x

∫ f(X) dX - a = 0

0

using Newton's method.

Big X represents the variable of f(X) to be integrated while small x is the x that needs to be solved for.

Taking the derivative of the above integral using the Second Fundamental Theorem of Calculus:

d/dx [ ∫( f(X) dX from X=0 to X=x ) - a ]

= d/dx [ F(x) - F(0) - a ]

= d/dx [ F(x) ] - d/dx [ F(0) ] - d/dx [ a ]

= d/dx [ F(x) ]

= f(x)

F(X) is the anti-derivative of f(X). F(0) and a are numerical constants, hence the derivative of each evaluates to 0.

Newton's Method to solve for any function g(x) is:

x_n+1 = x_n - g(x_n) / g'(x_n)

Applying this to the equation, Newton's Method gives:

x_n+1 = x_n - [ ∫( f(X) dX from X=0 to X=x_n ) - a ] / f(x_n)

**HP Prime Program INTEGRALSOLVE**

Note: Enter f(X) as a string and use capital X. This program is designed to be use in Home mode.

EXPORT INTEGRALSOLVE(f,a,x)

BEGIN

// f(X) as a string, area, guess

// ∫(f(X) dX,0,x) = a

// EWS 2019-07-26

// uses Function app

LOCAL x1,x2,s,i,w;

F0:=f;

s:=0;

x1:=x;

WHILE s==0 DO

i:=AREA(F0,0,x1)-a;

w:=F0(x1);

x2:=x1-i/w;

IF ABS(x1-x2)<1 font="" then="">

s:=1;

ELSE

x1:=x2;

END;

END;

RETURN approx(x2);

END;

**Examples**

Radians angle mode is set.

Example 1:

Solve for x:

x

∫ sin(X) dX = 0.75

0

Initial guess: 1

Result: x ≈ 1.31811607165

Example 2:

Solve for x:

x

∫ e^(X^2) dX = 0.95

0

Initial guess: 2

Result: x ≈ 0.768032819934

Pretty powerful.

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Eddie

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