What is f(x), if f '(x)=sinx/x and f(4)=4
f(x)=?+integral?
yes, but ∫ sinx/x dx cannot be done using elementary functions. It is something called Si(x), and is defined as that integral.
∫ sinx/x dx and others like ∫ cosx/x dx look innocent at first, but are some of the hardest integrals
check out the method in the video by "blackpenredpen"
https://www.youtube.com/watch?v=s1zhYD4x6mY
You can't integrate that elementarily
I used Taylor series expansion
Sinx contain all odds
Sinx=x-x³/3!+x⁵/5!-x⁷/7!+.......
Now sinx/x=x/x[1-x²/3!+x⁴/5!-x⁶/7!+x⁸/9!)
∫sinxdx/x=x-(x³/3.3!)+(x⁵/5.5)-(x⁷/(7.7!)+x⁹/(9.9!)...=f(x)
To find the function f(x) given f '(x) and an initial condition f(4)=4, we can use the process of integration.
First, let's integrate f '(x)=sin(x)/x.
The integral of sin(x)/x does not have a straightforward elementary function representation, but it can be expressed using a special function called the sine integral, denoted as Si(x).
So, integrating f '(x), we have: f(x) = Si(x) + C, where C is the constant of integration.
Now, we need to use the initial condition f(4)=4 to find the value of the constant C.
Plugging in x=4 into the equation f(x) = Si(x) + C, we get 4 = Si(4) + C.
To find the value of Si(4), you can use a calculator or reference table for the sine integral (Si) function.
Once you know the value of Si(4), you can solve the equation 4 = Si(4) + C for the constant C.
Finally, substitute the obtained value of C back into the equation f(x) = Si(x) + C to get the complete expression for the function f(x).