Question

Find the value of$\int {\dfrac{{dx}}{{\log {x^x}\left[ {{{\left( {\log x} \right)}^2} - 3\log x - 10} \right]}}}$.

Answer 1

Integration is a process of integrating an infinitesimally small segment over its range where integration in general referred to as summing up slices. Integration is also the opposite of differentiation. Integration is generally represented by $\int {{x^n}dx} = \dfrac{1}{{n + 1}}{x^{n + 1}}$ where $dx$ is the small quantities of $x$ which are being added.
The logarithmic function is the inverse function of the exponential function given by the formula${\log_b}a = c \Leftrightarrow {b^c} = \log a$, where b is the base of the logarithmic function.
Here, in this question logarithmic function, has been given in a different manner and so, logarithmic properties have been used. Integration of a polynomial can be solved in several ways, here we used the substitution method for solving the integration of $\int {\dfrac{{dx}}{{\log {x^x}\left[ {{{\left( {\log x} \right)}^2} - 3\log x - 10} \right]}}}$.
In the substitution method, a part under integration is to be considered as a parametric function, and the rest of the calculation is done with that parametric function only. At last, that parametric function has been replaced with the original value.

Complete step by step solution:
Consider, $u = \log x$
Differentiate both sides of the equation, $u = \log x$
$\dfrac{{du}}{{dx}} = \dfrac{d}{{dx}}(\log x) = \dfrac{1}{x} \\\ dx = xdu \\\$
The given equation can be written as:

$$I = \int {\dfrac{{dx}}{{\log {x^x}\left[ {{{\left( {\log x} \right)}^2} - 3\log x - 10} \right]}}} \\\ = \int {\dfrac{{dx}}{{x\log x\left[ {2\left( {\log x} \right) - 3\log x - 10} \right]}}} \\\ = \int {\dfrac{{xdu}}{{ux\left[ { - \log x - 10} \right]}}} \\\ = - \int {\dfrac{{du}}{{u(u + 10)}}} \\\$$

Also,

$$\dfrac{1}{{u(u + 10)}} = \dfrac{A}{u} + \dfrac{B}{{u + 10}} \\\ = \dfrac{{A(u + 10) + Bu}}{{u(u + 10)}} \\\ = \dfrac{{u(A + B) + 10A}}{{u(u + 10)}} \\\ 10A = 1{\text{ and, A + B = 0}} \\\ {\text{A = }}\dfrac{1}{{10}}{\text{ and, B = }}\dfrac{{ - 1}}{{10}} \\\$$ $$I = - \int {\dfrac{{du}}{{10u}}} + \int {\dfrac{{du}}{{10(u + 10)}}} \\\ = - \dfrac{{\log u}}{{10}} + \dfrac{{\log (u + 10)}}{{10}} + C \\\$$

Here, C is a constant of integration.
Now, substitute the value of u as $\log x$ to determine the integral in the variable of x only, we get:

$$I = - \dfrac{{\log u}}{{10}} + \dfrac{{\log (u + 10)}}{{10}} + C \\\ = \dfrac{{ - \log (\log x)}}{{10}} + \dfrac{{\log (\log x + 10)}}{{10}} + C \\\$$

Hence, $\int {\dfrac{{dx}}{{\log {x^x}\left[ {{{\left( {\log x} \right)}^2} - 3\log x - 10} \right]}}} = \dfrac{{ - \log (\log x)}}{{10}} + \dfrac{{\log (\log x + 10)}}{{10}} + C$

Additional Information: Some of the properties of logarithmic function are:
Power Rule: ${\log _b}{M^n} = n{\log _b}M$.
Product Rule: ${\log _a}(xy) = {\log _a}x + {\log _a}y$
Quotient Rule: ${\log _a}\left( {\dfrac{x}{y}} \right) = {\log _a}x - {\log _a}y$
Equality Rule: ${\log _a}x = {\log _a}y$ then, $x = y$.
Change of base rule: ${\log _a}x = \dfrac{{{{\log }_b}x}}{{{{\log }_b}a}}$

Note: It is interesting to note here that, the limits of the integration will also change according to the parametric values. Limits of the integration should always be dealt very carefully while using the substitution method for solving the integration.