Determine the antiderivative of 1/(x+5)

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neela | High School Teacher | (Level 3) Valedictorian

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To find the antiderivative of 1/(x+5).

Let f(x) = 1/(x+5).

To find the function F(x)  such that F'(x) = f(x) = 1/(x+5).

We know that d/dx { log (ax +b)} = (ax+b)'/(ax+b).

d/dx {log(ax+b)} = a/(ax+b).

Therefore equating a/(ax+b) = 1/(x+5) we get:

a(x+5) = ax+b

ax+5a = ax+b

ax = ax.

5a = b . Or b = 5a.

Therefore ax+b = ax+5a.

F(x) = log (ax+5a) = log a(x+5).

Or F(x) = log(x+5) +loga.

Therefore the anti derivative of 1/(x+5 ) is log (x+5) + a constant.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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