Integrate cos(4x)sin(x)

The easiest way of approaching this question is to use De Moivre's formula: e^(inx) = cos(nx) + isin(nx) from which it is simple to show that cos(nx) = (e^(inx) + e^(-inx)) / 2 and sin(nx) = (e^(inx))- e^(-inx)) /2i therefore, cos(4x)sin(x) = (e^(4ix) + e^(-4ix)) * ((e^(ix)) - (e^(-ix)) / 4i= [e^(5ix) - e^(-5ix) - e^(3ix) + e^(-3ix)] / 4i= sin(5x)/2 - sin(3x)/2Finally, integrating, this gives cos(3x)/6 - cos(5x)/10 + integration constantThis can also be done by using various trigonometric identities, however this method is simpler and can continue to be applied to more complex questions. 

KM
Answered by Kirill M. Further Mathematics tutor

13590 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

find the sum of r from 0 to n of : 1/((r+1)(r+2)(r+3))


find all the roots to the equation: z^3 = 1 + i in polar form


Solve the second order ODE, giving a general solution: x'' + 2x' - 3x = 2e^-t


Find the general solution for the determinant of a 3x3 martix. When does the inverse of this matrix not exist?


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences