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

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

Explanation

Solution

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\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.∫sechx tan⁡ hx dx=−sechx\int \text{sech} x\ \tan\ hx \ dx = -\text{sech} x
2.∫sech2x dx=tanh⁡x\int \text{sech}^{2}x\ dx = \tanh x

Complete step by step answer:
Given, sechx(tanh⁡x–sechx) dx\text{sech} x(\tanh x – \text{sech} x)\ dx
Let us consider the given hyperbolic function as II .
I=sechx(tanh⁡x–sechx) dxI = {\text{sech} x}(\tanh x – \text{sech} x)\ dx
By multiplying the terms inside and by removing the parentheses,
We get,
⇒ I=sechx tanh⁡x–sech2x dx\Rightarrow \ I = \text{sech} x\ \tanh x – \text{sech}^{2}{x\ dx}
On integrating,
We get,
I= ∫(sechx tanh⁡x–sech2x)dxI = \ \int\left( \text{sech} x\ \tanh x – \text{sech}^{2}x \right){dx}

By splitting the integral into two terms,
We get,
I= ∫sechx tanh⁡x − ∫sech2x dx I = \ \int \text{sech} x\ \tanh x\ - \ \int \text{sech}^{2}{x\ dx}\
We know that ∫sechx tan⁡ hx dx=−sechx\int \text{sech} x\ \tan\ hx \ dx = -\text{sech} x and ∫sech2x dx=tanh⁡x\int \text{sech}^{2}x\ dx = \tanh x
By using the integral rules,
We get,
I=−sechx–tanh⁡x+cI = -\text{sech} x – \tanh x + c
Where cc is the constant of integration.
On taking the minus sign common,
We get,
∴I=−(sechx+tanh⁡x)+c\therefore I = -(\text{sech} x + \tanh x) + c

Thus we get the integral of sechx(tanh⁡x–sechx) dx\text{sech} x(\tanh x – \text{sech} x)\ dx is −(sechx+tanh⁡x)+c-(\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)\sin(x) in trigonometry is cosh⁡(x)\cosh(x) and the integral of sinh⁡(x)\sinh(x) in hyperbolic functions is cosh⁡(x)\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 `cc’. cc is called the constant of integration. The variable xx in dx{dx} is known as the variable of integration or integrator.