The integral $$\int (\sin x + 2\cos x)^3\,dx$$ can be simplified using binomial expansion and trigonometric identities. The integrand can be expressed as a sum of terms involving only sine or cosine functions, allowing for straightforward integration. Utilizing de Moivre's formula to convert higher powers into multiple-angle sines or cosines further simplifies the integration process. Changing the variable to $$u = x + \alpha$$ enhances the ease of integration.
PREREQUISITESStudents and educators in calculus, particularly those focusing on integration techniques involving trigonometric functions.
Not sure that it helps, but you could rewrite the original integrand as ##A\sin^3(x+\alpha)##.Helly123 said:How to do this in simpler way?
That seems to be more or less the method embarked upon.andrewkirk said:First do a binomial expansion of the integrand. That will have only four terms.
The powers of the sine and cosine in each term will either be 3 and 0, or 2 and 1 (unordered pairs). Where they are 2 and 1, use the identity that sine squared plus cos squared equals 1 to turn the squared trig function into the other sort of trig function. Then you will have six terms, each of which is a power of only sine or only cosine.
Then use de Moivre's formula to convert the higher powers into multiple-angle sines or cosines with power 1.
Integration will then be straightforward.
haruspex said:Not sure that it helps, but you could rewrite the original integrand as ##A\sin^3(x+\alpha)##.
These terms can be easily integrated using a substitution. There's no need to use the Pythagorean identity here.andrewkirk said:Where they are 2 and 1, use the identity that sine squared plus cos squared equals 1 to turn the squared trig function into the other sort of trig function.