SF10

From Example Problems
Jump to: navigation, search

\int _{0}^{\infty }3^{{-4z^{2}}}dz\,

Rewrite the problem as

\int _{0}^{\infty }e^{{\ln \left[(3)^{{-4z^{2}}}\right]}}dz=\int _{0}^{\infty }e^{{-4z^{2}\ln 3}}dz\,

Let x=-4z^{2}\ln 3\, so the last integral is now

\int _{0}^{\infty }e^{{-x}}d\left({\sqrt  {{\frac  {x}{4\ln 3}}}}\right)=\int _{0}^{\infty }e^{{-x}}{\frac  {{\frac  {1}{2}}x^{{-{\frac  {1}{2}}}}}{2{\sqrt  {\ln 3}}}}dx={\frac  {1}{4{\sqrt  {\ln 3}}}}\int _{0}^{\infty }e^{{-x}}x^{{-{\frac  {1}{2}}}}dx\,

={\frac  {1}{4{\sqrt  {\ln 3}}}}\Gamma \left({\frac  {1}{2}}\right)={\sqrt  {{\frac  {\pi }{16\ln 3}}}}\,

Special Functions

Calculus

Main Page