Given the derivative of a function, how would you find the original function?
I am trying to teach myself calculus, so I am asking this question. For example, given that the derivative of a function is 2x, you could use the d/dx [x^n] = (n)(x(n-1)) to guess that the original function was x2, but what would you do if you were told that x2 itself was the derivative of another function? Please help me. Note that I am NOT someone asking for homework help, my homework from school is not calculus.
1
answer
|
Answer it!
|
It is called the anti-derivative, also known as an indefinite integral. So:
The integral of (xn)= (xn+1)/(n+1))
So in the case of x2. You would have, (x2+1)/(2+1) which is (x3)/3 + C. I added a (C) because C is any integer, and when the derivative is taken of an integer, it would = 0. So then when you check, the derivative of (x3)/3 , it would equal x2.
This concept does get more challenging, when you have definite integrals, radicals, or when you need to integrate through substitution.
The integral of (xn)= (xn+1)/(n+1))
So in the case of x2. You would have, (x2+1)/(2+1) which is (x3)/3 + C. I added a (C) because C is any integer, and when the derivative is taken of an integer, it would = 0. So then when you check, the derivative of (x3)/3 , it would equal x2.
This concept does get more challenging, when you have definite integrals, radicals, or when you need to integrate through substitution.
 |
