H2 Math Integration by Parts: Area Under a Logarithmic Curve
With Mr Gan3 min 42 secPublished
Watch Mr Gan work through 2022 H2 Math Paper 1 Question 7: differentiate a logarithmic function, sketch its graph and use integration by parts to find an exact area.
What you will learn
- Differentiate (ln x)/x³ using the quotient rule or product rule.
- Identify the intercept, maximum point and horizontal asymptote.
- Use integration by parts to evaluate an exact area.
Lesson notes
A written summary of the method, with mathematical notation clarified for revision.
Read the graph before integrating
For y = (ln x)/x³ with x > 0, the derivative is (1 − 3 ln x)/x⁴. Setting it to zero gives the stationary point at x = e^(1/3), with y = 1/(3e).
The curve crosses the x-axis at x = 1 and approaches y = 0 as x increases. Choose a calculator viewing window that shows the maximum clearly, rather than copying a graph whose important features are compressed.
Find the exact area from x = 1 to x = 3
The curve is above the x-axis on this interval, so the area is the integral of (ln x)/x³ from 1 to 3.
For integration by parts, let u = ln x and dv/dx = x⁻³. Then du/dx = 1/x and v = −1/(2x²). An antiderivative is −(ln x)/(2x²) − 1/(4x²).
Substituting the limits gives the exact area 2/9 − (ln 3)/18. Keep the logarithm in the answer when an exact value is required.
Try it yourself
Differentiate the antiderivative before substituting the limits. Then explain why the lower limit is 1 rather than 0.
Check your reasoning
Differentiation returns (ln x)/x³. The lower limit is the curve's x-intercept, x = 1; the function is not defined at x = 0.