Question

How do you find the integral of $\cos (5x)$?

Answer 1

Hint: In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, its inverse operation, differentiation, is the other.

Complete step by step answer:
To solve this question,
Let $5x = t$
$ \Rightarrow dt = 5dx$
Therefore,
$\begin{gathered}
  \int {\cos \left( {5x} \right)dx} \\
   \Rightarrow \int {\cos t\dfrac{{dt}}{5}} \\
   \Rightarrow \dfrac{{\sin t}}{5} + c \\
  \dfrac{{\sin (5x)}}{5} + c \\
\end{gathered} $
Now, look carefully and observe that $\cos $ in the given question results to be changed in $\sin $ as the end result of this question.

Note: The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. For this reason, the terms integral may be also referring to the related notion of the antiderivative, called an indefinite integral.