Moderate -0.3 This is a straightforward 'show that' integration by parts question with a simple polynomial factor (1-x) and exponential function. The definite integral has convenient limits (0 to 1) and requires only one application of integration by parts followed by routine evaluation. Slightly easier than average due to the direct nature and verification format.
Attempt integration by parts and reach \(k(1-x)e^{-\frac{x}{2}} \pm k\int e^{-\frac{x}{2}}dx\), or equivalent
M1
Obtain \(-2(1-x)e^{-\frac{x}{2}} - 2\int e^{-\frac{x}{2}}dx\), or equivalent
A1
Integrate and obtain \(-2(1-x)e^{-\frac{x}{2}} + 4e^{-\frac{x}{2}}\), or equivalent
A1
Use limits \(x = 0\) and \(x = 1\), having integrated twice
M1
Obtain the given answer correctly
A1
[5]
Attempt integration by parts and reach $k(1-x)e^{-\frac{x}{2}} \pm k\int e^{-\frac{x}{2}}dx$, or equivalent | M1 |
Obtain $-2(1-x)e^{-\frac{x}{2}} - 2\int e^{-\frac{x}{2}}dx$, or equivalent | A1 |
Integrate and obtain $-2(1-x)e^{-\frac{x}{2}} + 4e^{-\frac{x}{2}}$, or equivalent | A1 |
Use limits $x = 0$ and $x = 1$, having integrated twice | M1 |
Obtain the given answer correctly | A1 | [5]