AntiderivativesFind the antiderivative of 1/(x+5)

2 Answers

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

Int dx/(x+5)

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.