**HP Prime and TI Nspire CX CAS: Solving Integral Equations**

**Introduction**

The program INTEGRALSOLVE solve the following equation:

(Format: ∫( integrand dvar, lower, upper)

∫( f(t) dt, 0, x) = a

∫( f(t) dt, 0, x) – a = 0

It is assumed that x>0.

We can use the Second Theorem of Calculus which takes the derivative of the integral:

d/dx ∫( f(t) dt, a, x) = f(x)

We don’t have to worry about lower limit a at all for the theorem to work.

∫( f(t) dt, 0, x) – a

Take the derivative with respect to x on both sides (d/dx):

= d/dx ∫( f(t) dt, 0, x) – a

= d/dx ∫( f(t) dt, 0, x) – d/dx a

Let F(t) be the anti-derivative of f(t):

= d/dx (F(x) – F(0)) – 0

= d/dx F(x) – d/dx F(0)

F(0) is a constant.

= f(x)

Newton’s Method to find the roots of f(x) can be found by the iteration:

x_(n+1) = x_n – f(x_n) / f'(x_n)

Applying that to find the roots of ∫( f(t) dt, 0, x) – a:

x_(n+1) = x_n – (∫( f(t) dt, 0, x_n) – a) / f(x_n)

**HP Prime Program INTEGRALSOLVE**

Enter f(X) as a string as it will be stored in Function App variable F0. Use X as the independent variable.

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)

s:=1;

ELSE

x1:=x2;

END;

END;

RETURN approx(x2);

END;

**TI NSpire CX CAS Program INTEGRALSOLVE**

(Caution: This program needs to be typed in)

Use t as the independent variable.

Define LibPub integralsolve(f,a,x)=

Func

:© f(x), area, guess: ∫(f(t) dt,0,x = a)

:Local x1,x2,s

:s:=0

:x1:=x

:While s=0

: x2:=x1-((∫(f,t,0,x1)-a)/(f|t=x1))

: If abs(x2-x1)≤1E−12 Then

: s:=1

: Else

: x1:=x2

: EndIf

:EndWhile

:Return approx(x2)

:EndFunc

**Examples**

Example 1:

∫( 2*t^3 dt, 0, x) = 16

Guess = 2

Root ≈ 2.3784

Example 2:

∫( sin^2 t dt, 0, x) = 1.4897

Guess = 1

(Radians Mode)

Root ≈ 2.4999

Source:

Green, Larry. “The Second Fundamental Theorem of Calculus” Differential Calculus for Engineering and other Hard Sciences. Lake Tahoe Community College. http://www.ltcconline.net/greenl/courses/105/Antiderivatives/SECFUND.HTM Retrieved July 25, 2019

Happy Solving!

Eddie

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