Antiderivative of ln x

The antiderivative of ln x is equal to the difference of x ln x and x plus the constant of integration. Mathematically, we can write the antiderivative of ln x as ∫ ln x dx = x ln x - x + C, where C is the integration constant. Since integration is nothing but the reverse process of differentiation, therefore integral of ln x is the same as the antiderivative of ln x.

Further in this article, we will evaluate the antiderivative of ln x using the method of integration by parts and determine its formula. We will also determine the antiderivative of ln x square and ln x by x along with some solved examples for a better understanding of the concept.

The antiderivative of ln x is the integral of the natural logarithmic function and is given by x ln x - x + C, where C is the constant of integration. To find the antiderivative of ln x, we need to determine the value of ∫ln x dx, where the integration is with respect to the variable x. The integration of ln x is mathematically written as ∫ln x dx = x ln x - x + C. This antiderivative of ln x can be calculated using one of the important methods of integration called the method of integration by parts. Now, let us go through its formula given in the next section.

As we know that the antiderivative of ln x is equal to x ln x - x + C, hence its formula is written as ∫ln x dx = x ln x - x + C, where dx shows the integration is with respect to x, ∫ is the symbol of integration and dx shows the antiderivative of ln x is w.r.t. to x. The image given below shows the formula for the integral of ln x:

Now that we know the antiderivative of ln x, we will prove its formula using the method of integration by parts. To evaluate ∫ ln x dx, we will use integration by parts formula ∫u dv = uv − ∫vdu. Assume u = ln(x) and dv = dx ⇒ v = x. We will use the formula for the derivative of ln x given by, d(ln x)/dx = 1/x. Therefore, we have

∫ln(x) dx = x ln(x) − ∫x.(ln(x))′ dx

⇒ ∫ln(x) dx = x ln(x) − ∫x × (1/x) dx [Because d(ln x)/dx = 1/x]

⇒ ∫ln(x) dx = x ln(x) − ∫dx

⇒ ∫ln(x) dx = x ln(x) − x + C

Hence, we have the antiderivative of ln x which is given by, ∫ln(x) dx = x ln(x) − x + C

In this section, we will find the antiderivative of ln x square, that is, [ln x]² using the antiderivative of ln x. We will evaluate the antiderivative of [ln x]² using the method of integration by parts ∫u dv = uv − ∫vdu. For this, assume u = [ln x]² and dv = dx. This implies we have v = x and du = 2 (ln x)/x. Therefore, we have

∫ [ln x]² dx = x [ln x]² - ∫x [2 (ln x)/x] dx

= x [ln x]² - 2 ∫ln x dx

= x [ln x]² - 2(x ln x - x + C) [Because the antiderivative of ln x is ∫ln(x) dx = x ln(x) − x + C]

= x [ln x]² - 2x ln x + 2x + 2C

= x [ln x]² - 2x ln x + 2x + K, where K = 2C is the integration constant.

Hence, the antiderivative of ln x square is given by ∫ [ln x]² dx = x [ln x]² - 2x ln x + 2x + K.

Important Notes on Antiderivative of ln x

• The antiderivative of ln x is the integral of the natural logarithmic function and is given by x ln x - x + C.
• The antiderivative of ln x can be calculated using the method of integration by parts.
• ∫ [ln x]² dx = x [ln x]² - 2x ln x + 2x + K

☛ Related Topics:

• Log Formulas
• Integral Calculus
• Applications of Integrals

What is the Antiderivative of ln x in Calculus?

The antiderivative of ln x is the integral of the natural logarithmic function and is given by x ln x - x + C, where C is the constant of integration. Mathematically, we can write the antiderivative of ln x as ∫ ln x dx = x ln x - x + C, where C is the integration constant.

How to Find the Antiderivative of ln x?

We can find the antiderivative of ln x using one of the common methods of integration. It can be determined using the method of integration by parts and the formula that we use is ∫u dv = uv − ∫vdu, where we can assume u = ln x and dv = dx.

What is the Antiderivative of (ln x)²?

The antiderivative of ln x square is ∫ [ln x]² dx = x [ln x]² - 2x ln x + 2x + K, where K is the constant of integration. We can determine this antiderivative using the method of integration by parts ∫u dv = uv − ∫vdu.

Is Antiderivative of ln x the Same as the Integral of ln x?

Since the integration of a function is nothing but the process of reverse differentiation of the function, therefore we can say that the antiderivative of ln x is the same as the integral of ln x.

What is the Antiderivative of ln x by x?

The antiderivative of ln x by x is given by the formula ∫(1/x) ln x dx = (ln x)²/2 + C which can be evaluated using the substitution method.