Question

How do you evaluate the integral of $\int {{{(\ln x)}^3}} dx$?

Answer 1

The above question is based on the concept of integration. Since it is an indefinite integral which has no upper and lower limits, we can apply integration properties by integrating it in parts so that we can find the antiderivative of the above equation of natural log.

Complete step by step solution:
Integration is a way of finding the antiderivative of any function. It is the inverse of differentiation. It denotes the summation of discrete data. Calculation of small problems is an easy task but for adding big problems which include higher limits, integration methods are used. The above given equation is an indefinite integral which means there are no upper or lower limits given.

The above equation should be in the below form.
$\int {f(x) = F(x) + C}$
where C is constant.

So, the first step is that we will be to substitute the term ln in another variable.
Let
$t = \ln x$

Therefore,

$$x = {e^t} \\\ dx = {e^t}dx \\\$$

By substituting in the original equation,
$\int {{{(\ln x)}^3}dx = \int {{t^3}{e^t}dt} }$

For integrating the equation in parts, we get the formula,
$\int {f(x)g'(x)dx = f(x)g(x) - \int {f'(x)g(x)dx} }$

where f(x) and g(x) are two functions and $g'(x)$ and $f'(x)$ is the derivative of the functions.

Now, integrating in parts we get,

$$\Rightarrow \int {{t^3}{e^t}dt = \int {{t^3}d({e^t}) = {t^3}{e^t} - 3\int {{t^2}{e^t}dt} } } \\\ \Rightarrow \int {{t^2}{e^t}dt = \int {{t^2}d({e^t}) = {t^2}{e^t} - 2\int {t{e^t}dt} } } \\\ \Rightarrow \int {t{e^t}dt = \int {td({e^t}) = t{e^4} - \int {{e^t}dt = t{e^t} - {e^t} + C} } } \\\$$

Substituting all of them,
$\int {{t^3}{e^t}dt = {t^3}{e^t} - 3{t^2}{e^t} + 6t{e^t} - 6{e^t} + C = {e^t}({t^3} - 3{t^2} + 6t - 6) + C}$

Now substitute in original equation,
$\int {{{(\ln x)}^3}dx = x({{(\ln x)}^3} - 3{{(\ln x)}^2} + 6\ln x - 6) + C}$

Note: An important thing to note is that ln is also called a natural log. It is the inverse of e. Basically,the natural logarithm of a number is its logarithm to the base of mathematical constant e whereas log has a common base of 10.