AntiderivativesFind the antiderivative of 1/(x+5)
You need to evaluate the indefinite integral of the function `y = 1/(x+5)` such that:
`int 1/(x+5) dx`
Using the following formula yields:
`int 1/(x+a) dx = ln|x+a| + c`
Reasoning by analogy yields:
`int 1/(x+5) dx = ln|x + 5| + c`
Hence, evaluating the indefinite integral yields `int 1/(x+5) dx = ln|x + 5| + c.`
We'll calculate the antiderivative F(x) integrating the given function.
F(x) is a function such as dF/dx = f(x).
Int f(x)dx = F(x) + C
We'll substitute the denominator of the function by another variable:
x + 5 = t
We'll differentiate both sides:
(x+5)'dx = dt
dx = dt
We'll re-write the indefinite integral using the variable t:
Int dt/t = ln |t| + C
We'll substitute t by the expression in x:
F(x) = ln |x+5| + C
The antiderivative of the function y = 1/(x+5) is F(x)=ln |x+5| + C.