STEP I 2011 2

Write:

\displaystyle \frac {xe^x} {1 + x} = \frac {(x + 1 - 1) e^x} {1 + x} = e^x - \frac {e^x} {1 + x}

So we have:

\begin{align*}\int_0^1 \frac {xe^x} {1 + x} & = \int_0^1 e^x \mathrm dx - \int_0^1 \frac {e^x} {1 + x} \\ & = [e^x]_0^1 - E \\ & = e - 1 - E\end{align*}

For the second integral we have:

\displaystyle \frac {x^2} {1 + x} = \frac {x^2 - 1} {1 + x} + \frac 1 {1 + x} = \frac {(x - 1)(x + 1)} {1 + x} + \frac 1 {1 + x} = (x - 1) + \frac 1 {1 + x}

So:

\begin{align*}\int_0^1 \frac {x^2 e^x} {1 + x} \mathrm dx & = \int_0^1 (x - 1)e^x \mathrm dx + \int_0^1 \frac {e^x} {1 + x} \mathrm dx \\ & = \int_0^1 (x - 1)e^x \mathrm dx + E \\ & = [(x - 1)e^x]_0^1 - \int_0^1 e^x \mathrm dx + E \\ & = 1 - [e^x]_0^1 + E \\ & = 1 - [e - 1] + E \\ & = 2 + E - e \end{align*}

using IBP.

(i)

With a view to make the integral resemble the integral for E or the other two integrals we’ve worked out let:

\displaystyle u = \frac {1 - x} {1 + x} = -\frac {x - 1} {x + 1} = -\left(1 - \frac 2 {x + 1}\right) = -1 + \frac 2 {x + 1}

Differentiating:

\displaystyle \frac {\mathrm du} {\mathrm dx} = -\frac 2 {(x + 1)^2}

So \displaystyle \mathrm dx = -\frac {(x + 1)^2} 2 \mathrm du. So:

\begin{align*}\int_0^1 \frac {e^{\frac {1 - x} {1 + x}}} {1 + x} \mathrm dx & = -\frac 1 2 \int_1^0 \frac {e^u (x + 1)^2} {(1 + x)} \mathrm du \\ & = \frac 1 2 \int_0^1 e^u (x + 1) \mathrm du\end{align*}

Note that:

u + 1 = \dfrac 2 {x + 1}

So:

\displaystyle x + 1 = \dfrac 2 {u + 1}

Meaning that the integral is equal to:

\displaystyle \frac 2 2 \int_0^1 \frac {e^u} {1 + u} \mathrm du = E

(ii)

With similar motivation to before, let u = x^2, then \displaystyle \mathrm dx = \frac 1 {2x} \mathrm du. Then:

\begin{align*}\int_1^{\sqrt 2} \frac {e^{x^2}} x \mathrm dx & = \frac 1 2 \int_1^2 \frac {e^u} {x^2} \mathrm du \\ & = \frac 1 2 \int_1^2 \frac {e^u} u \mathrm du \\ & = \frac 1 2 \int_0^1 \frac {e^{u + 1}} {u + 1} \mathrm du \\ & = \frac e 2 \int_0^1 \frac {e^u} {1 + u} \mathrm du \\ & = \frac {Ee} 2\end{align*}

1 Like

:clap: :clap: