Integrate the following integral:
\[\int {\sin mx\cos nx dx} ,m \ne n\].
Answer
365.1k+ views
Hint:Break the given integral into two parts using trigonometric identities
We have the given integral as
\[\int {\sin mx\cos nxdx} \]
Now, divide and multiply by 2,
\[ = \dfrac{1}{2}\int {2\sin mx\cos nxdx} \]
We know the identity
\[2\sin A\cos B = \sin (A + B) + \sin (A - B)\]
Using this identity, such that $A = mx$and $B = nx$ we get,
$ = \dfrac{1}{2}\int {\sin (mx + nx) + \sin (mx - nx)} dx$
$ = \dfrac{1}{2}\int {\sin (m + n)x + \sin (m - n)x} dx$
We know that $\int {\sin ax} dx = \dfrac{{ - \cos ax}}{a}$
Therefore, we get,
\[ = \dfrac{1}{2}\left( {\dfrac{{ - \cos (m + n)}}{{m + n}} - \dfrac{{\cos (m - n)}}{{m - n}}} \right) + c\]
Where $c$ is an integration constant.
Note: To solve these types of questions, we must have an adequate knowledge of various integration properties and identities, evaluating the integral within such parameters, will lead towards the required solution.
We have the given integral as
\[\int {\sin mx\cos nxdx} \]
Now, divide and multiply by 2,
\[ = \dfrac{1}{2}\int {2\sin mx\cos nxdx} \]
We know the identity
\[2\sin A\cos B = \sin (A + B) + \sin (A - B)\]
Using this identity, such that $A = mx$and $B = nx$ we get,
$ = \dfrac{1}{2}\int {\sin (mx + nx) + \sin (mx - nx)} dx$
$ = \dfrac{1}{2}\int {\sin (m + n)x + \sin (m - n)x} dx$
We know that $\int {\sin ax} dx = \dfrac{{ - \cos ax}}{a}$
Therefore, we get,
\[ = \dfrac{1}{2}\left( {\dfrac{{ - \cos (m + n)}}{{m + n}} - \dfrac{{\cos (m - n)}}{{m - n}}} \right) + c\]
Where $c$ is an integration constant.
Note: To solve these types of questions, we must have an adequate knowledge of various integration properties and identities, evaluating the integral within such parameters, will lead towards the required solution.
Last updated date: 20th Sep 2023
•
Total views: 365.1k
•
Views today: 5.65k
Recently Updated Pages
What is the Full Form of DNA and RNA

What are the Difference Between Acute and Chronic Disease

Difference Between Communicable and Non-Communicable

What is Nutrition Explain Diff Type of Nutrition ?

What is the Function of Digestive Enzymes

What is the Full Form of 1.DPT 2.DDT 3.BCG
