Question

What is the integral of $\operatorname{Sin}2x$?

Answer 1

To integrate the trigonometric function we will use the basic concept of integrating the simple trigonometric function, like$\sin \left( ax+b \right)$where ‘a’ and ‘b’ are constant and ‘x’ is variable.

Complete step-by-step solution:
Moving ahead with the question in step wise manner,
We are asked to integrate$\operatorname{Sin}2x$. Since we directly know the integration of$\sin x$ which is$-\cos x$. But integration of$\operatorname{Sin}2x$ is a little bit complex. For this type of question we know that we can integrate the trigonometric function like integration of$\sin \left( ax+b \right)$ (where ‘a’ and ‘b’ are constant and ‘x’ is variable) which is equal$\dfrac{-\cos \left( ax+b \right)}{a}+c$, i.e.$\int{\sin \left( ax+b \right)}=\dfrac{-\cos \left( ax+b \right)}{a}+c$ .
So using the same formula let us integrate$\operatorname{Sin}2x$. By comparing it with the formula we can say it is written as$\operatorname{Sin}\left( 2x+0 \right)$in which constant ‘a’ and ‘b’ are ‘2’ and ‘0’ respectively.
So according to question we had to find out the integration of$\operatorname{Sin}\left( 2x+0 \right)$which we can write it as$\int{\operatorname{Sin}\left( 2x+0 \right)}$.
As by formula we know that integration of trigonometric function$\sin x$is$-\cos x$so by using the formula of$\sin \left( ax+b \right)$we will get;
$\begin{aligned} & \int{\sin 2x=}\int{\operatorname{Sin}\left( 2x+0 \right)} \\\ & \int{\sin 2x=}\dfrac{-\cos \left( 2x+0 \right)}{2}+c \\\ & \int{\sin 2x=}\dfrac{-\cos \left( 2x \right)}{2}+c \\\ \end{aligned}$
So we got$\dfrac{-\cos \left( 2x+0 \right)}{2}+c$in which c is some constant.
Hence the answer is$\dfrac{-\cos \left( 2x+0 \right)}{2}+c$.

Note: If the angle inside the trigonometric function is present in linear form then only this formula as in our case it is$2x$and if it will be$2{{x}^{2}}$or${{x}^{3}}$or some other exponential then we can’t apply this method.