Question 3 on the OCR MEI C3 June 2015 paper. Find the exact value of Integral x^3 ln x dx between 1 and 2.

This is a good candidate for integration by parts because it is a product of two functions – ln(x) and x^3, and x^3 is easy to integrate. The definition for integration by parts is: integral(u dv/dx) dx = uv – integral(v du/dx) dx. We choose u = ln(x) as ln(x) is difficult to integrate and dv/dx = x^3. This means v = (x^4)/4 and du/dx = 1/x. Integral(ln(x) x^3) dx = ln(x) * (x^4)/4 – integral(((x^4)/4 )* 1/x) dx = ln(x) * (x^4)/4 - integral((x^3)/4 ) dx = ln(x) * (x^4)/4 – (x^4)/16. Now we must substitute in the bounds. [ln(2) * (2^4)/4 - (2^4)/16] - [ln(1) * (1^4)/4 - (1^4)/16 ]. Remember that ln(1) = 0 giving an answer of ln(2) * 16/4 - 16/16 + 1/16 = 4 ln(2) – 15/16.

Answered by Tom K. Maths tutor

7687 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

A curve C is defined by the parametric equations x=(4-e^(2-6t))/4 , y=e^(3t)/(3t), t doesnt = 0. Find the exact value of dy/dx at the point on C where t=2/3 .


The random variable J has a Poisson distribution with mean 4. Find P(J>2)


How can we determine stationary points by completing the square?


What are radians, why can't we just use degrees?


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