Exam Board OCR MEI Module C3 (Core Mathematics 3) Year 2016 Session June Marks 5 Paper Download PDF ↗ Mark scheme Download PDF ↗ Topic Integration by Parts Type Integration of x^n·ln(x) Difficulty Challenging +1.8 This question requires integration by parts with a non-standard integrand (x^{-1/2} ln x), careful handling of the logarithm and fractional power, and evaluation over bounds that include a potential singularity concern at x=0 (though the interval [-1,4] appears to have a typo since ln x is undefined for negative x). The combination of techniques and need for algebraic manipulation of surds makes this significantly harder than routine C3 integration questions. Spec 1.08i Integration by parts
Find \(\int_{-1}^4 x^{-\frac{1}{2}} \ln x dx\), giving your answer in an exact form. [5]
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Answer: let \(u = \ln x\), \(u' = 1/x\), \(v' = x^{-1/2}\), \(v = kx^{1/2}\)Marks: M1Guidance: soi \((k \neq 0)\)Answer: \(\int x^{-1/2} \ln x \, dx = [2x^{1/2} \ln x] - \int 2x^{1/2} \cdot \frac{1}{x} dx\)Marks: A1Guidance: (none)Answer: \(= [2x^{1/2} \ln x] - \int 2x^{-1/2} dx\)Marks: M1Guidance: \(x^{1/2}/x = x^{-1/2}\) or \(1/x^{1/2}\) seenAnswer: \(= [2x^{1/2} \ln x - 4x^{1/2}]^4_1\)Marks: A1Guidance: May be integrated separatelyAnswer: \(= 4 \ln 4 - 8 - (2\ln 1 - 4)\)Marks: A1Guidance: (none)Answer: \(= 4 \ln 4 - 4\)Marks: A1caoGuidance: oe (eg ln 256−4) but must evaluate ln1=0; mark final answer
Total Marks: [5]
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**Answer:** let $u = \ln x$, $u' = 1/x$, $v' = x^{-1/2}$, $v = kx^{1/2}$ | **Marks:** M1 | **Guidance:** soi $(k \neq 0)$
**Answer:** $\int x^{-1/2} \ln x \, dx = [2x^{1/2} \ln x] - \int 2x^{1/2} \cdot \frac{1}{x} dx$ | **Marks:** A1 | **Guidance:** (none)
**Answer:** $= [2x^{1/2} \ln x] - \int 2x^{-1/2} dx$ | **Marks:** M1 | **Guidance:** $x^{1/2}/x = x^{-1/2}$ or $1/x^{1/2}$ seen
**Answer:** $= [2x^{1/2} \ln x - 4x^{1/2}]^4_1$ | **Marks:** A1 | **Guidance:** May be integrated separately
**Answer:** $= 4 \ln 4 - 8 - (2\ln 1 - 4)$ | **Marks:** A1 | **Guidance:** (none)
**Answer:** $= 4 \ln 4 - 4$ | **Marks:** A1cao | **Guidance:** oe (eg ln 256−4) but must evaluate ln1=0; mark final answer
**Total Marks:** [5]
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Find $\int_{-1}^4 x^{-\frac{1}{2}} \ln x dx$, giving your answer in an exact form. [5]
\hfill \mbox{\textit{OCR MEI C3 2016 Q3 [5]}}