3 Find the exact value of
$$\int _ { 1 } ^ { 4 } x ^ { \frac { 3 } { 2 } } \ln x \mathrm {~d} x$$
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Question 3:
Answer Marks
Guidance
Answer Mark
Guidance
Commence integration and reach \(ax^{\frac{5}{2}} \ln x + b\int x^{\frac{5}{2}} \cdot \frac{1}{x}\,dx\) M1*
Obtain \(\frac{2}{5}x^{\frac{5}{2}} \ln x - \frac{2}{5}\int x^{\frac{5}{2}} \cdot \frac{1}{x}\,dx\) A1
Complete the integration and obtain \(\frac{2}{5}x^{\frac{5}{2}} \ln x - \frac{4}{25}x^{\frac{5}{2}}\), or equivalent A1
Use limits correctly, having integrated twice e.g. \(\frac{2}{5}\times 32\ln 4 - \frac{4}{25}\times 32 - \left(\frac{2}{5}\times 0\right) + \frac{4}{25}\) DM1
Obtain answer \(\frac{128}{5}\ln 2 - \frac{124}{25}\), or exact equivalent A1
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## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| Commence integration and reach $ax^{\frac{5}{2}} \ln x + b\int x^{\frac{5}{2}} \cdot \frac{1}{x}\,dx$ | M1* | |
| Obtain $\frac{2}{5}x^{\frac{5}{2}} \ln x - \frac{2}{5}\int x^{\frac{5}{2}} \cdot \frac{1}{x}\,dx$ | A1 | |
| Complete the integration and obtain $\frac{2}{5}x^{\frac{5}{2}} \ln x - \frac{4}{25}x^{\frac{5}{2}}$, or equivalent | A1 | |
| Use limits correctly, having integrated twice e.g. $\frac{2}{5}\times 32\ln 4 - \frac{4}{25}\times 32 - \left(\frac{2}{5}\times 0\right) + \frac{4}{25}$ | DM1 | |
| Obtain answer $\frac{128}{5}\ln 2 - \frac{124}{25}$, or exact equivalent | A1 | |
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3 Find the exact value of
$$\int _ { 1 } ^ { 4 } x ^ { \frac { 3 } { 2 } } \ln x \mathrm {~d} x$$
\hfill \mbox{\textit{CAIE P3 2020 Q3 [5]}}