Question

How do you integrate $ \cos (5x)dx? $

Answer 1

Hint: As we can see here there are two functions involved simultaneously in this question, one is cos and another is $ 5x $ so we have to substitute $ 5x $ as u, and while substituting we also have to substitute dx in the equation.

Complete step by step solution:
Given, $ \int {\cos (5x)dx = ?} $
Let this equation be I, so we can write this as,
 $ I = \int {\cos (5x)dx} $
Now as said, let us substitute $ 5x $ as u, and also calculate dx in term of u
 $ u = 5x $
 $ du = 5(dx) $
Rearranging this we can have our value of dx as,
 $ dx = \dfrac{{du}}{5} $
Substituting these values in equation I we will get it as,
 $ I = \int {\cos (u)\dfrac{{du}}{5}} $
As $ \dfrac{1}{5} $ is constant so we can take it from integration, so our equation will become,
\[I = \dfrac{1}{5}\int {\cos (u)du} \]
Now we have to only calculate the integral value of cos (u), i.e.
 $ \int {\cos (u)} = \sin (u) + c $
Using this result, we can write our equation as,
 $ I = \dfrac{1}{5}\int {\cos (u)du} = \dfrac{1}{5}\sin (u) + c $
Here, as we all know ‘c’ is the integration constant.
Now substituting back the value of u in our equation we will get,
 $ \dfrac{1}{5}\sin (u) + c = \dfrac{1}{5}\sin (5x) $
So our final result can be written as,
 $ I = \int {\cos (5x)dx} = \dfrac{1}{5}\sin (5x) + c $
So, the correct answer is “ $ \dfrac{1}{5}\sin (5x) + c $ ”.

Note: The result should be written in the form of a given variable, so after calculating the result one should substitute back the value of variable as per the substitution failure to which results will not be in most desirable form.