Question

Evaluate $\int {\left( {\dfrac{m}{x} + \dfrac{x}{m} + {m^x}} \right)} dx$

Answer 1

Hint : We have to find the integration of $\int {\left( {\dfrac{m}{x} + \dfrac{x}{m} + {m^x}} \right)} dx$ . Here m is the constant and x is a variable. Divide the integral into 3 parts separated with addition. $\int {\dfrac{1}{x}dx = \ln x;\int {xdx = \dfrac{{{x^2}}}{2};\int {{a^x}dx = \dfrac{1}{{\ln a}}{a^x}} } }$ Use these formulas to solve the above integral.

Complete step-by-step answer :
We are given $\int {\left( {\dfrac{m}{x} + \dfrac{x}{m} + {m^x}} \right)} dx$ and we have to find it’s integral.
Here, in the given integral m is a constant and x is a variable, m is a constant because we need to find the integration with respect to x. So m should be considered as a constant.
Divide the integral into three parts.
$\int {\left( {\dfrac{m}{x} + \dfrac{x}{m} + {m^x}} \right)} dx = \int {\dfrac{m}{x}dx + \int {\dfrac{x}{m}dx + \int {{m^x}dx} } }$
Solve for each part separately and integrate it.
$\int {\dfrac{m}{x}dx = \int {m\left( {\dfrac{1}{x}} \right)dx} } \\\ = m\int {\dfrac{1}{x}dx} \\\ = m\left( {\ln x} \right) \\\ = m{\log _e}x \\\ \int {\dfrac{x}{m}dx = \int {\dfrac{1}{m}\left( x \right)dx} } \\\ = \dfrac{1}{m}\int {xdx} \\\ = \dfrac{1}{m}\left( {\dfrac{{{x^2}}}{2}} \right) \\\ \left( {\because \int {{x^n} = \dfrac{{{x^{n + 1}}}}{{n + 1}}} } \right) \\\ = \dfrac{{{x^2}}}{{2m}} \\\ \int {{m^x}dx = \dfrac{1}{{\ln m}}\left( {{m^x}} \right)} \\\ = \dfrac{{{m^x}}}{{{{\log }_e}m}} \\\$
$\int {\left( {\dfrac{m}{x} + \dfrac{x}{m} + {m^x}} \right)} dx = m{\log _e}x + \dfrac{{{x^2}}}{{2m}} + \dfrac{{{m^x}}}{{{{\log }_e}m}}$
ln a=log a with base e

Note : Integration is the opposite of differentiation. An algebraic expression used to determine the change incurred from one point to another with a unit is referred to as differentiation. On the other hand, integration is an algebraic expression used in calculating the area under the curve. We use the derivative to determine the maximum and minimum values of particular functions. Integration is mainly used in technology to improve data accessibility, better communication, and robust growth. Differentiation represents the rate of change of a function. Integration represents an accumulation or sum of a function over a range. Differentiation and Integration both can have limits. They both are literally inverses. So, do not confuse differentiation with integration.