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

Sep 2, 2014

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:

$F \left(x\right) = 4 {x}^{2} + 3 x + C$

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

$f \left(x\right) = {x}^{n}$ then $f ' \left(x\right) = n {x}^{n - 1}$

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

$F \left(x\right) = \frac{{x}^{n + 1}}{n + 1}$

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:

$f \left(x\right) = 6 {x}^{3} + 5$ and $g \left(x\right) = 6 {x}^{3} + 25$

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