Question

How do you evaluate the integral `int8x+3 dx`?

Answer 1

When taking integrals, you will normally solve them one term at a time. You will do the inverse of the power rule so the answer would be:

`F(x) = 4x^2 + 3x + C`

Integrals are the inverse of derivatives so you follow the rules in reverse. The `8x` can be written as `8x^1`. To take the derivative of this you would multiply the coefficient by one then subtract one from the exponent, so if:

`f(x) = x^n` then `f'(x)=nx^(n-1)`

To reverse the power rule, you will first add one to the exponent then divide the whole term by the new term:

`F(x) = (x^(n+1))/(n+1)`

Both terms in this problem can be solved with the power rule.

Due to this being a indefinite integral, not having any bounds, you will have to put `+ C` do to the possibility of a constant being dropped when a derivative was taken. In other words:

`f(x) = 6x^3 + 5` and `g(x) = 6x^3 + 25`

would have the same derivative because the constant becomes zero and the additive identity property states that anything added to zero is unchanged.