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Question: Integrate the following integral: \[\int {\sin mx\cos nx dx} ,m \ne n\]....

Integrate the following integral:
∫sin⁡mxcos⁡nxdx,m≠n\int {\sin mx\cos nx dx} ,m \ne n.

Explanation

Solution

Hint:Break the given integral into two parts using trigonometric identities

We have the given integral as
∫sin⁡mxcos⁡nxdx\int {\sin mx\cos nxdx}
Now, divide and multiply by 2,
=12∫2sin⁡mxcos⁡nxdx= \dfrac{1}{2}\int {2\sin mx\cos nxdx}
We know the identity
2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B)2\sin A\cos B = \sin (A + B) + \sin (A - B)
Using this identity, such that A=mxA = mxand B=nxB = nx we get,
=12∫sin⁡(mx+nx)+sin⁡(mx−nx)dx= \dfrac{1}{2}\int {\sin (mx + nx) + \sin (mx - nx)} dx
=12∫sin⁡(m+n)x+sin⁡(m−n)xdx= \dfrac{1}{2}\int {\sin (m + n)x + \sin (m - n)x} dx
We know that ∫sin⁡axdx=−cos⁡axa\int {\sin ax} dx = \dfrac{{ - \cos ax}}{a}
Therefore, we get,
=12(−cos⁡(m+n)m+n−cos⁡(m−n)m−n)+c= \dfrac{1}{2}\left( {\dfrac{{ - \cos (m + n)}}{{m + n}} - \dfrac{{\cos (m - n)}}{{m - n}}} \right) + c
Where cc 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.