how to find the antiderivative of lnx
How to Find the Antiderivative of Ln(x)
Though it may be basic, computing the antiderivative of ln(x) requires the use of some important, fundamental concepts of calculus. In this Buzzle article, we tell you what these concepts are, and how you can use them to find the antiderivative of ln(x).
Did You Know?
The word 'calculus' is a Latin word, which means a small stone used for counting. Today, calculus refers to any method or system of calculation that comprises symbolic manipulation of the different expressions.Integral of Ln(x) by Parts
For finding out the antiderivative of ln(x), you need to first understand the concept of integration by parts. The following is a step-by-step derivation of the formula for integration by parts.
Step 1
We will start by using the product rule of derivatives. Consider that there are two functions - f(x) and g(x), and we have to find the derivative of the multiplication of those two. This is represented as following.
((f(x) · g(x))' = f(x) · g'(x) + f'(x) · g(x) -------------(1)
Step 2
Now, we will integrate both the sides of equation (1).
∫ ((f(x) · g(x))' = ∫ { f(x) · g'(x) + f'(x) · g(x) }
f(x) · g(x) = ∫ f(x) · g'(x) dx + ∫ f'(x) · g(x) dx ---------------- (2)
Note: Here, we don't need to include an integration constant on the left side of the equation, as the integrals on the right side already have integration constants.
Step 3
Rearranging the terms in equation (2), we get,
∫ f(x) · g'(x) dx = f(x) · g(x) - ∫ f'(x) · g(x) dx
The equation obtained in Step 3 is the formula for computing integration by parts. It is frequently used for solving difficult integration problems.
To make the integration by parts formula easier to remember, the following substitutions are used.
f(x) = u
g(x)'dx = dv
f'(x)dx = du
g(x) = v
Thus, the formula becomes,
∫ udv = uv - ∫ vdu
To integrate by parts, keep in mind to choose 'u' and 'dv' strategically, so that the calculation becomes easy.How to Find the Antiderivative of Ln(x)
Now that we are clear with the concept of integration by parts, let us see how we can use it to find the antiderivative that is, the integral of ln(x)dx.
∫ ln(x) dx
To begin with, we first need to choose u and dv. On first inspection, it might appear that there is only one term in the expression above. However, since any number multiplied by 1 yields the same number again, we can substitute the following-
∫ ln(x) · 1dx
Now, let us make the following two assignments.
u = ln(x) ----------(a)
and,
dv = 1 · dx -----------(b)
Taking the derivative of u, we get,
du = (1/x) dx ------------(c)
and taking the integration of du, we get,
∫ du = ∫ dv
u = x ------------(d)
We have the formula for integration by parts as,
∫ udv = uv - ∫ vdu
Substituting values from equations (a), (b), (c), and (d) into the above formula for integration by parts, we get,
∫ ln(x) dx = ln(x) · x dx - ∫ 1/x · x dx
= x * ln(x) - ∫ dx
= x * ln(x) - x + C
The above expression is the antiderivative or the integration of the natural logarithm of x. To verify our answer, let us take its derivative to see if we get back the function ln(x).
[x * ln(x) - x + C]' = ln(x) + (x/x) - 1 + 0
= ln(x) + 1 - 1 + 0
= ln(x)