7 Use the substitution \(u = 3 - x ^ { 3 }\) to find the exact value of \(\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { 3 - x ^ { 3 } } \mathrm {~d} x\).
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Question 7:
\(\int_0^1 \frac{x^5}{3-x^3} \, dx\), substitution \(u = 3 - x^3\)
Answer Marks
Guidance
Working/Answer Mark
Guidance
\(\frac{du}{dx} = -3x^2 \Rightarrow x^2 \, dx = -\frac{du}{3}\) M1
Correct differentiation
\(x^3 = 3 - u\), so \(x^5 = x^3 \cdot x^2 = (3-u)x^2\) M1
Attempt to express \(x^5\) in terms of \(u\)
Limits: \(x=0 \Rightarrow u=3\); \(x=1 \Rightarrow u=2\) B1
Both limits correct
\(\int_3^2 \frac{(3-u)}{u}\cdot\left(-\frac{1}{3}\right) du = \frac{1}{3}\int_2^3 \frac{3-u}{u} \, du\) A1
Correct transformed integral
\(= \frac{1}{3}\int_2^3 \left(\frac{3}{u} - 1\right) du\) M1
Splitting fraction
\(= \frac{1}{3}\left[3\ln u - u\right]_2^3\) A1
Correct integration
\(= \frac{1}{3}\left[(3\ln 3 - 3) - (3\ln 2 - 2)\right]\) \(= \frac{1}{3}\left[3\ln\frac{3}{2} - 1\right] = \ln\frac{3}{2} - \frac{1}{3}\) A1
Exact final answer
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# Question 7:
$\int_0^1 \frac{x^5}{3-x^3} \, dx$, substitution $u = 3 - x^3$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{du}{dx} = -3x^2 \Rightarrow x^2 \, dx = -\frac{du}{3}$ | M1 | Correct differentiation |
| $x^3 = 3 - u$, so $x^5 = x^3 \cdot x^2 = (3-u)x^2$ | M1 | Attempt to express $x^5$ in terms of $u$ |
| Limits: $x=0 \Rightarrow u=3$; $x=1 \Rightarrow u=2$ | B1 | Both limits correct |
| $\int_3^2 \frac{(3-u)}{u}\cdot\left(-\frac{1}{3}\right) du = \frac{1}{3}\int_2^3 \frac{3-u}{u} \, du$ | A1 | Correct transformed integral |
| $= \frac{1}{3}\int_2^3 \left(\frac{3}{u} - 1\right) du$ | M1 | Splitting fraction |
| $= \frac{1}{3}\left[3\ln u - u\right]_2^3$ | A1 | Correct integration |
| $= \frac{1}{3}\left[(3\ln 3 - 3) - (3\ln 2 - 2)\right]$ | | |
| $= \frac{1}{3}\left[3\ln\frac{3}{2} - 1\right] = \ln\frac{3}{2} - \frac{1}{3}$ | A1 | Exact final answer |
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7 Use the substitution $u = 3 - x ^ { 3 }$ to find the exact value of $\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { 3 - x ^ { 3 } } \mathrm {~d} x$.\\[0pt]
[6 marks]
\hfill \mbox{\textit{AQA C3 2014 Q7 [6]}}