| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Integration of x^n·ln(x) |
| Difficulty | Standard +0.3 This is a straightforward integration by parts question with a standard form (x^n·ln(x)). Students need to recognize the technique, apply it correctly with u=ln(x) and dv=x²dx, then evaluate the definite integral. While it requires careful execution and simplification to exact form, it's a textbook example with no novel insight required, making it slightly easier than average. |
| Spec | 1.08i Integration by parts |
Evaluate $\int_1^2 x^2 \ln x \, dx$, giving your answer in an exact form. [5]
M1: Use integration by parts with $u = \ln x$, $dv = x^2 \, dx$
M1: Correctly find $du = \frac{1}{x} \, dx$ and $v = \frac{x^3}{3}$
M1: Write $\int x^2 \ln x \, dx = \frac{x^3}{3} \ln x - \int \frac{x^3}{3} \cdot \frac{1}{x} \, dx = \frac{x^3}{3} \ln x - \int \frac{x^2}{3} \, dx$
M1: Integrate to get $\frac{x^3}{3} \ln x - \frac{x^3}{9}$
A1: Evaluate between limits: $\left[\frac{8}{3} \ln 2 - \frac{8}{9}\right] - \left[0 - \frac{1}{9}\right] = \frac{8}{3} \ln 2 - \frac{7}{9}$
---2 Evaluate $\int _ { 1 } ^ { 2 } x ^ { 2 } \ln x \mathrm {~d} x$, giving your answer in an exact form.
\hfill \mbox{\textit{OCR MEI C3 2007 Q2 [5]}}