How do you find the antiderivative of #4/sqrtx#?
The first step is to recognize that
When we take the derivative of an exponential term, we bring down the power (i.e. multiply the term you are deriving by the value of its exponent) and reduce the power by one. When we take an antiderivative of an exponential term, we want to do the opposite. In essence, we are "undoing" the derivative. Thus, we need to increase the power by one, and multiply the term by the reciprocal of that final exponent.
Because this function we want to "anti-derive" has a negative exponent, we need to make sure we pay a little extra attention. The exponent is
However, we can see that our original function does not have that
Because this would be an indefinite integral (i.e. having no specific values for which its variables should be defined), we need to account for any constants that could have been present when the derivative was taken, since when we take the derivative of a constant, it reduces to 0. How could we know that the antiderivative is
We can account for this by adding
Our final answer, therefore, is: