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Question: Solve that\[\int {\sin 4x.\cos 3x{\kern 1pt} {\kern 1pt} {\kern 1pt} dx} \]...

Solve that∫sin⁡4x.cos⁡3xdx\int {\sin 4x.\cos 3x{\kern 1pt} {\kern 1pt} {\kern 1pt} dx}

Explanation

Solution

Integration is also known as the antiderivative in mathematics. In this equation we are going to integrate sin⁡4xcos⁡3x\sin 4x\cos 3x . First we are going to expand using the formula 2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B)2\sin A\cos B = \sin (A + B) + \sin (A - B) and after expanding, we are going to use the formula∫sin⁡(ax)dx=−cos⁡(ax)a\int {\sin (ax){\kern 1pt} dx = \dfrac{{ - \cos (ax)}}{a}} to integrate. We will attain a value for the given integral.

Complete answer:
We write the given trigonometric function as a function f(x)f(x)i.e. f(x)=sin⁡4x.cos⁡3xf(x) = \sin 4x.\cos 3x and then we find its antiderivative or its integral with respect toxx.
That is \int {f(x)} {\kern 1pt} {\kern 1pt} dx$$$$ = $$$$\int {\sin 4x.\cos 3x{\kern 1pt} {\kern 1pt} {\kern 1pt} dx}
From our previous knowledge let us use the formula,
2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B)2\sin A\cos B = \sin (A + B) + \sin (A - B)
Then let us covert the integral into,
12∫2sin⁡4x.cos⁡3xdx\dfrac{1}{2}\int {2\sin 4x.\cos 3x{\kern 1pt} {\kern 1pt} {\kern 1pt} dx}
Now using the formula above,
=12∫sin⁡(4x+3x)+sin⁡(4x−3x)dx= {\kern 1pt} {\kern 1pt} {\kern 1pt} \dfrac{1}{2}\int {\sin (4x + 3x) + \sin (4x - 3x){\kern 1pt} dx}
=12∫sin⁡(7x)+sin⁡(x)dx= {\kern 1pt} {\kern 1pt} {\kern 1pt} \dfrac{1}{2}\int {\sin (7x) + \sin (x){\kern 1pt} dx}
Now we know that ∫sin⁡(ax)dx=−cos⁡(ax)a\int {\sin (ax){\kern 1pt} dx = \dfrac{{ - \cos (ax)}}{a}}
Therefore, the integration is,
=12[−cos⁡7x7−cos⁡x]+C= {\kern 1pt} {\kern 1pt} {\kern 1pt} \dfrac{1}{2}[\dfrac{{ - \cos 7x}}{7} - \cos x] + C
=−cos⁡7x14−12cos⁡x+C= \dfrac{{ - \cos 7x}}{{14}} - {\kern 1pt} {\kern 1pt} {\kern 1pt} \dfrac{1}{2}\cos x + C
Hence the integration of the function is
−cos⁡7x14−12cos⁡x+C\dfrac{{ - \cos 7x}}{{14}} - {\kern 1pt} {\kern 1pt} {\kern 1pt} \dfrac{1}{2}\cos x + C

Additional information:
Differentiation and integration is taken into consideration by means of all scientists in the course as one of the first-class sciences that guided the thoughts of guy over all times. The fields of using calculus are very extensive. It enters into many fields and is not restricted to specific human beings or to folks that use it best but to nearly all humans. Calculus is a part of arithmetic and is likewise utilized in physics. With calculus, we can find how the converting situations of a gadget have an effect on us. You can discover ways to control a system by means of analyzing calculus. Calculus is the language of engineers, scientists, and economists. From your microwaves, mobile phones, TV, and vehicle to medicinal drug, economy, and country wide protection all need calculus and so right here is how calculus is used in our each day lives.

Note:
It is important that we know the basic differentiation and integration of trigonometric functions, it is also very important that we know how to convert one function to another of whose integration we know beforehand so as to get the solution easily.