Question

What is the integral of $\cos \left( {2\theta } \right)$ with respect to $\theta$ ?

Answer 1

Hint : The given question requires us to integrate a function of $\theta$ with respect to $\theta$ . Integration gives us a family of curves. Integrals in math are used to find many useful quantities such as areas, volumes, displacement, etc. integral is always found with respect to some variable, which in this case is $\theta$ .

Complete step-by-step answer :
The given question requires us to integrate a trigonometric function $\cos \left( {2\theta } \right)$ in variable $\theta$ . So, we can integrate the given function by substituting $2\theta$ as x.
So, we have,
$\int {\cos \left( {2\theta } \right)} \,d\theta$
So, we substitute $2\theta$ as x.
We have, $x = 2\theta$ . Differentiating both sides, we get,
$\Rightarrow dx = 2d\theta$
$\Rightarrow d\theta = \dfrac{{dx}}{2}$
So substituting the value of $d\theta$ in terms of dx, we get,
$\Rightarrow \int {\cos \left( x \right)} \,\dfrac{{dx}}{2}$
Now, we know that the integral of cosine is sine. So, we get,
$\Rightarrow \dfrac{1}{2}\sin \left( x \right) + c$
Now, substituting back the value of x in terms of $\theta$ , we get,
$\Rightarrow \dfrac{1}{2}\sin \left( {2\theta } \right) + c$
So, the integral of the function $\cos \left( {2\theta } \right)$ with respect to $\theta$ is $\dfrac{1}{2}\sin \left( {2\theta } \right) + c$ where c is the arbitrary constant of indefinite integration.
So, the correct answer is “ $\dfrac{1}{2}\sin \left( {2\theta } \right) + c$ ”.

Note : The indefinite integrals of certain functions may have more than one answer in different forms. However, all these forms are correct and interchangeable into one another. Indefinite integral gives us the family of curves as we don’t know the exact value of the constant.