Integrate ⌠( xcos^2(x))dx

We must first use trigonometric identities to simplify cos2(x). We can use the formula cos(A+B) = cos(A)cos(B) - sin(A)sin(B) , where A=x and B=x, so that we get cos(2x) = cos2(x) - sin2(x) = 2cos2(x) - 1. Rearranging this we find that cos2(x) = 1/2 + (cos(2x))/2 This gives us (⌠(x+xcos(2x))dx)/2 = 1/2(⌠( x)dx + ⌠( xcos(2x))dx) = x2 /4 + ⌠( xcos(2x))dx) /2 We can then use the integration by parts formula, ⌠(udv)dx = uv - f(vdu)dx , where u=x and dv=cos(2x), so that we get x2 /4 + ⌠( xcos(2x))dx) /2 = x2 /4 + xsin(2x)/4 - ⌠(sin(2x))dx)/4 = x2 /4 + xsin(2x)/4 + cos(2x)/8 Hence,the final answer is x2 /4 + xsin(2x)/4 + cos(2x)/8 + c

Answered by Daniel A. Maths tutor

9814 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

For y = 7x - x^3, find the two stationary points and what type of stationary points they are.


Let f(x)=e^x sin(x^2). Find f'(x)


A curve has the equation y=sin(x)cos(x), find the gradient of this curve when x = pi. (4 marks)


Differentiate y=(sin(x))^(2)


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences