Question

Find the integration of $\int {\left( {\sin x + \cos x} \right)} dx$

Answer 1

We are given a sum of two trigonometric functions and are asked to integrate them and since the operation used is addition we can split them and using the formulae $\int {\sin x} dx = - \cos x$ and $\int {\cos x} dx = \sin x$ we get the required solution.

Complete step-by-step answer:
We are asked to integrate the given function $\int {\left( {\sin x + \cos x} \right)} dx$
Here we can see that the operation between the two trigonometric functions is addition
Hence we can split this into two parts
That is ,
$\Rightarrow \int {\left( {\sin x + \cos x} \right)} dx = \int {\sin xdx} + \int {\cos xdx}$
Now we need to make use of our basic integration formulas
When we integrate $\sin x$ with respect to x we get $- \cos x$
And when we integrate $\cos x$ with respect to x we get $\sin x$
Now substituting this we get
$\Rightarrow \int {\left( {\sin x + \cos x} \right)} dx = - \cos x + \sin x + C \\\ \Rightarrow \int {\left( {\sin x + \cos x} \right)} dx = sin x - cos x + C$
And here C is a arbitrary constant

Note: We split the function inside the integral only because the operation between them is addition.
If the operation is multiplication then we cannot split them we need to apply integration by parts.
Many students tend to get confused with the integration formula of $\cos x$ and $\sin x$.