Question

Evaluate the iterated integrand \(\int_0^1 {\int_0^v {\sqrt {1 + {e^v}} } } dwdv. \)

Short answer

Value of integral is \(\frac{2}{3}\left( {{{(1 + e)}^{\frac{3}{2}}} - {2^{\frac{3}{2}}}} \right)\).

Step-by-step solution

Integrating on \(w\) First.

\(\int_0^1 {\int_0^{{e^v}} {\sqrt {1 + {e^v}} } } dwdv = \left. {\int_0^1 w \sqrt {1 + {e^v}} } \right|_0^{{e^v}}dv\)

\( = \int_0^1 {{e^v}} \sqrt {1 + {e^v}} dv\).

Integrate on \(V\).\(\)

\(\int_0^1 {{e^v}} \sqrt {1 + {e^v}} dv = \int_2^{1 + e} {{u^{1/2}}} du\)

\( = {\left. {\frac{2}{3}{u^{3/2}}} \right|_2}1 + e\)

\( = \frac{2}{3}\left( {{{(1 + e)}^{3/2}} - {2^{3/2}}} \right).\)