#MathUnusual | How can Integrate ?

integration sine 5x by 2 divided by sine x by 2 dx. How can we solve this problem? Can we simplify the function? No, because it is already in simplified form. Can we use any trigonometric formula? No, because we don't have any trigonometric formula for sine 5x by 2 sine x by 2. Can we convert into form of any trigonometric formula? Yes, but how can? Just multiply and divide by 2 cos x by 2. Then what we have? In the numerator we have 2 sine 5x by 2 cos x by 2 and in the denominator 2 sine x by 2 into cos x by 2. So for the numerator we have trigonometric formula 2 sine alpha cos beta equal to sine alpha plus beta sine alpha minus beta and for the denominator we have 2 sine alpha cos alpha equal to sine 2 alpha. So if you apply this formula then we have in the numerator we have sine 5x by 2 plus x by 2 plus sine 5x by 2 minus x by 2 divided by here alpha equal to x by 2 so 2 times x by 2 equal to x so in the denominator we have sine x then we have sine 3x plus sine 2x divided by sine x. We have trigonometric formula for sine 2 alpha sine 3 alpha for sine 2 alpha it is 2 sine alpha cos alpha and sine 3 alpha equal to 3 sine alpha minus 4 sine 3 alpha. Now substitute that value in the integration we can see in the numerator and denominator sine x is a common factor so if we cancel that common factor from the numerator and denominator we have sine 3x is now 3 4 sine cube x now it's 4 sine square x 2 sine x cos x is now 2 cos x we can see that 3 can be easily integrate cos x is also easily integrate but sine square x we cannot easily integrate. So we can use another trigonometric formula for sine square x sine square alpha equal to 1 minus cos 2 alpha divided by 2. If we substitute then we have 4 divided by 2 equal to 2 sine square x equal to 1 minus cos 2x then after simplification of this term 3 minus 2 equal to 1 minus minus plus 2 cos 2x plus 2 cos x. Now we can integrate all the term easily and we have x plus integration of cos 2x is equal to sine 2x by 2 so 2 2 cancel we have sine 2x plus 2 times sine x and integration constant c. Thanks for watching for more videos like, share, subscribe, press the bell icon of next gen mess.