Question

How do you integrate $ \int \dfrac{{\sin x}}{{{{(2 + 3\cos x)}^2}}}dx $ using substitution?

Answer 1

Hint: To solve this question, first we will assume any of the set of variables or constant be another variable to get the expression easier. And conclude until the non-operational state is not achieved. And finally substitute the assumed value. Here we put the denominator part as u and solve further.

Complete step by step solution:
The given expression: $ \int \dfrac{{\sin x}}{{{{(2 + 3\cos x)}^2}}}dx $
Let $ u = 2 + 3\cos x $
Differentiate the above equation that we supposed:
 $ \Rightarrow du = - 3\sin xdx $
 $ \Rightarrow \sin xdx = - \dfrac{1}{3}du $
Now, put the upper values in the main given expression:
 $
 \int \dfrac{{\sin x}}{{{{(2 + 3\cos x)}^2}}}dx \\
   = - \dfrac{1}{3}\int \dfrac{{du}}{{{u^2}}} \\
   = - \dfrac{1}{3}\int {u^{ - 2}}du \\
   = \dfrac{1}{3}.\dfrac{1}{u} \\
   = \dfrac{1}{3}.(\dfrac{1}{{2 + 3\cos x}}) + C \;
  $
So, the correct answer is “$ \dfrac{1}{3}.(\dfrac{1}{{2 + 3\cos x}}) + C$”.

Note: Usually the method of integration by substitution is extremely useful when we make a substitution for a function whose derivative is also present in the integrand. Doing so, the function simplifies and then the basic formulas of integration can be used to integrate the function.