Standard +0.3 This is a straightforward integration by parts question with a standard form (x^n·ln(x)). The technique is routine for P3 level: let u=ln(x) and dv=x^(-3/2)dx, integrate by parts, then evaluate definite integral limits. The fractional power and logarithm evaluation require care but follow standard procedures, making it slightly easier than average.
Integrate by parts and reach \(ax^{-\frac{1}{2}}\ln x + b\int x^{-\frac{1}{2}}\cdot\frac{1}{x}\,dx\)
M1*
Obtain \(-2x^{-\frac{1}{2}}\ln x + 2\int x^{-\frac{3}{2}}\cdot\frac{1}{x}\,dx\), or equivalent
A1
Complete the integration, obtaining \(-2x^{-\frac{1}{2}}\ln x - 4x^{-\frac{1}{2}}\), or equivalent
A1
Substitute limits correctly, having integrated twice
M1(dep*)
Obtain the given answer following full and correct working
A1
## Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate by parts and reach $ax^{-\frac{1}{2}}\ln x + b\int x^{-\frac{1}{2}}\cdot\frac{1}{x}\,dx$ | M1* | |
| Obtain $-2x^{-\frac{1}{2}}\ln x + 2\int x^{-\frac{3}{2}}\cdot\frac{1}{x}\,dx$, or equivalent | A1 | |
| Complete the integration, obtaining $-2x^{-\frac{1}{2}}\ln x - 4x^{-\frac{1}{2}}$, or equivalent | A1 | |
| Substitute limits correctly, having integrated twice | M1(dep*) | |
| Obtain the given answer following full and correct working | A1 | |