Question

Evaluate the integral of $\text{cosec}\ x*\cot\ x$ all divided by $2 - \text{cosec x}\ dx$?

Answer 1

In this question, we need to find the integral of $\text{cosec}\ x*\cot\ x$ all divided by $2 - \text{cosec x}\ dx$ . Integration is nothing but its derivative is equal to its original function. The inverse of differentiation is known as integral. The symbol ‘$\int$’ is known as the sign of integration and is used in the process of integration . First, we need to convert the given question into the form of mathematical expression. Then we can consider the given expression as $I$ . Then we can use the reverse chain rule by substituting the function as $u$ . After integrating, then we need to do some rearrangements of terms and hence we can find the integral of the given expression.

Complete step by step solution:
Reverse chain rule :
Reverse chain rule is also known as u-Substitution. U sub is a special method of integration. It is applicable, when the expression contains two functions. This method combines two functions with the help of another variable ‘u’ and makes the integration process direct and much easier.
$\int\ f(x)\ f’(x)\ dx$
Here the original component $f(x)$ and the derivative component $f’(x)\ dx$
$\int\ f(x)\ f’(x)\ dx = \int u\ du$
Given $\text{cosec}(x)*\cot(x)$ all divide by $2 - \text{cosec}(x)\ dx$
Now let us convert the given into the form of mathematical expression .
$\Rightarrow \dfrac{\text{cosec}\left( x \right)\cot\left( x \right)}{2 - \text{cosec}(x)}{dx}$
Here we need to find the integral of $\dfrac{\text{cosec}\left( x \right)\cot\left( x \right)}{2 - \text{cosec}(x)}{dx}$
Let us consider
$I = \int\dfrac{\text{cosec}\left( x \right)\cot\left( x \right)}{2 – \text{ cosec}(x)}{dx}$ ••• (1)
By using reverse chain rule,
Let us consider $u = 2 - \text{cosec}(x)$
On differentiating,
We get,
$\dfrac{du}{dx} = 0 – ( - \cot\left( x \right)\text{cosec}\left( x \right))$
On simplifying,
We get,
$\Rightarrow \ du = \cot(x)\ \text{cosec}(x)\ dx$
Thus the equation (1) becomes,
$I = \int\dfrac{1}{u}{du}$
On integrating,
We get,
$I = \log(|u|) + c$
Where $c$ is the constant of integration.
By substituting the value of $u = 2 - \text{cosec}(x)$
We get,
$I = \log(|2 - \text{cosec}(x)|) + c$
Thus we get the integral of $\text{cosec}(x)\ *\cot(x)$ all divide by $2 - \text{cosec}(x)\ dx$ is $\log(|2 - \text{cosec}\left( x \right)|) + c$.
The integral of $\text{cosec}(x)\ *\cot(x)$ all divide by $2 - \text{cosec}(x)\ dx$ is $\log(|2 - \text{cosec}\left( x \right)|) + c$.

Note:
The concept used in this question is integration method, that is integration by u-substitution .We should never forget to include the modulus symbol inside the logarithmic function in the answer and also we should not forget to add the constant of integration while solving problems related to indefinite integrals. Since this is an indefinite integral we have to add an arbitrary constant ‘$c$’. $c$ is called the constant of integration. The reverse chain rule method is related to the chain rule of differentiation, which when applied to antiderivatives is known as the reverse chain rule that is integration by u substitution.