Question

How do you integrate \[\text{sech} x(\tanh x – \text{sech} x)\ dx\] ?

Answer 1

Hint:In this given question, we need to integrate the given hyperbolic function. Hyperbolic functions are nothing but they are similar to Trigonometric functions. The representation of hyperbolic function is \[\sinh x\ ,\cosh x\] etc… . Integration is nothing but its derivative is equal to its original function. Integration is also known as antiderivative. The inverse of differentiation is known as integral. The symbol `\[\int\]’ is the sign of the integration. The process of finding the integral of the given function is known as integration. First we can consider the given function as I . Then we can split the given hyperbolic function into two terms. And by using the integral rules, we can integrate the given hyperbolic function

Formula used:
1.\[\int \text{sech} x\ \tan\ hx \ dx = -\text{sech} x \]
2.\[\int \text{sech}^{2}x\ dx = \tanh x \]

Complete step by step answer:
Given, \[\text{sech} x(\tanh x – \text{sech} x)\ dx\]
Let us consider the given hyperbolic function as \[I\] .
\[I = {\text{sech} x}(\tanh x – \text{sech} x)\ dx\]
By multiplying the terms inside and by removing the parentheses,
We get,
\[\Rightarrow \ I = \text{sech} x\ \tanh x – \text{sech}^{2}{x\ dx}\]
On integrating,
We get,
\[I = \ \int\left( \text{sech} x\ \tanh x – \text{sech}^{2}x \right){dx}\]

By splitting the integral into two terms,
We get,
\[I = \ \int \text{sech} x\ \tanh x\ - \ \int \text{sech}^{2}{x\ dx}\ \]
We know that \[\int \text{sech} x\ \tan\ hx \ dx = -\text{sech} x \] and \[\int \text{sech}^{2}x\ dx = \tanh x \]
By using the integral rules,
We get,
\[I = -\text{sech} x – \tanh x + c\]
Where \[c\] is the constant of integration.
On taking the minus sign common,
We get,
\[\therefore I = -(\text{sech} x + \tanh x) + c\]

Thus we get the integral of \[\text{sech} x(\tanh x – \text{sech} x)\ dx\] is \[-(\text{sech} x + \tanh x )+ c\].

Note:Mathematically the difference between the derivatives of Trigonometric functions and hyperbolic functions are the integral of \[\sin(x)\] in trigonometry is \[\cosh(x)\] and the integral of \[\sinh(x)\] in hyperbolic functions is \[\cosh(x)\] . The anti-derivative of the function is also known as the inverse of the derivative of the function . The concept used in this question is integration method, that is integration of the hyperbolic function . Since this is an indefinite integral we have to add an arbitrary constant `\[c\]’. \[c\] is called the constant of integration. The variable \[x\] in \[{dx}\] is known as the variable of integration or integrator.