Question

Evaluate:
\[\int {\left( {{x^m} + {m^x} + {m^m}} \right)} dx\]

Answer 1

Hint: The process integration is to find the antiderivative of a function. Also, integration is the inverse process of differentiation and is known as the anti-differentiation. The integrals are of two types, definite integral and indefinite integral. A definite integral contains upper and lower limits whereas an indefinite integral does not contain upper and lower limits.
Here, we are given an indefinite integral and we are asked to calculate the value of\[\int {\left( {{x^m} + {m^x} + {m^m}} \right)} dx\]

Complete step-by-step solution:
Some formulae that we need to apply in the solution are as follows.
$\int {dx = x + C} $
$\int {{x^n}} dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C$ ,$n \ne - 1$
\[\int {{a^x}dx = \dfrac{{{a^x}}}{{\ln a}} + C} \] ;$a > 0$,$a \ne 1$
Where \[C\] is any constant.
Complete step by step solution:
The given function is
\[\int {\left( {{x^m} + {m^x} + {m^m}} \right)} dx\]
The addition rule of indefinite integral states that the sum of two functions is the sum of the integrals of two functions. This can be applied for more than two functions too.
So, we need to apply the additional rule.
\[\int {\left( {{x^m} + {m^x} + {m^m}} \right)} dx\]$ = \int {{x^m}dx + } \int {{m^x}} dx + \int {{m^m}} dx$ ………………$\left( 1 \right)$
Let us consider
\[\int {{x^m}dx} \]
Now, we shall apply the formula
 $\int {{x^n}} dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C$
Hence,
\[\int {{x^m}dx} = \dfrac{{{x^{m + 1}}}}{{m + 1}} + {C_1}\]
Where ${C_1}$ is any constant.
Now, let us consider
$\int {{m^x}} dx$
Now, we shall apply the formula
 \[\int {{a^x}dx = \dfrac{{{a^x}}}{{\ln a}} + C} \]
Hence, we get
\[\int {{m^x}} dx = \dfrac{{{m^x}}}{{\ln m}} + {C_2}\]
Where ${C_2}$ is any constant.
Now, let us consider
$\int {{m^m}} dx$
Here, we shall take the constant outside.
$\int {{m^m}} dx = {m^m}\int {dx} $
Now, we shall apply the formula
 $\int {dx = x + C} $
Hence, we get
\[{m^m}\int {dx} = {m^m}x + {C_3}\]
Where ${C_3}$ is any constant.
Now, substitute all the values in the equation$\left( 1 \right)$
\[\int {\left( {{x^m} + {m^x} + {m^m}} \right)} dx\]$ = \int {{x^m}dx + } \int {{m^x}} dx + \int {{m^m}} dx$
\[ = \dfrac{{{x^{m + 1}}}}{{m + 1}} + \dfrac{{{m^x}}}{{\ln m}} + {m^m}x + C\]
Where
$C = {C_1} + {C_2} + {C_3}$ is any constant.

Note: We all know that differentiation is the process of finding the derivation of the functions whereas process integration is to find the antiderivative of a function. Hence, these two processes are said to be inverse to each other. That is integration is the inverse process of differentiation and known as the anti-differentiation.