The way to integrate a differentiated function(微分した関数を積分する方法)

I would like to know the way to integrate a differentiated function.

Specifically, I want a calculation like this.

y=x^2+x+5

diff(y)=2x+1

integral(diff(y) ,0 ,1)=2

I tried some ways, but they were not success.

I show the ways I tried.

ex.1

>> y=@(x) x.^2+x+5

y =

  function_handle with value:

    @(x)x.^2+x+5

>> diff(y)
Undefined function 'diff' for input arguments of type 'function_handle'.

ex.2

>> syms x;
>> y=x.^2+x+5
 
y =
 
x^2 + x + 5
 
>> diff(y)
 
ans =
 
2*x + 1
 
>> yy=@(x) diff(y)

yy =

  function_handle with value:

    @(x)diff(y)

>> integral(yy,0,1)
Error using integralCalc/finalInputChecks (line 522)
Input function must return 'double' or 'single' values. Found 'sym'.

Error in integralCalc/iterateScalarValued (line 315)
                finalInputChecks(x,fx);

Error in integralCalc/vadapt (line 132)
            [q,errbnd] = iterateScalarValued(u,tinterval,pathlen);

Error in integralCalc (line 75)
        [q,errbnd] = vadapt(@AtoBInvTransform,interval);

Error in integral (line 88)
Q = integralCalc(fun,a,b,opstruct);
 

Do you have any ways? Please teach me...

Thank you.

Answer 1

[Daniel Pollard]

Well the integral of a derivative is simply the function itself. For example:

y(x) = x^2 + x + 5

dy/dx = 2x + 1

integral(dy/dx) = x^2 + x + 5 + c

integral(dy/dx from 0 to 1) = y(1) - y(0).

In your case, you have already defined y(x), so you can just use that.

[Yoshiki Sato]

Thank you for your support.

I know that you said.

Actually, I would like to use two differentiated functions, y1'(x) and y2'(x).

And, I want to integrate the function defined by y1'(x) * y2'(x) .

To explain simply, I used simple example.

Thank you.

[Daniel Pollard]

I haven't used the symbolic toolbox before so this is getting a bit out of my knowledge. Do you know the functions beforehand, so maybe you could integrate y1'*y2' using integration by parts, or explicitely define the product in matlab and use a symbolic integration command?