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.