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Genrish500 [490]
3 years ago
9

exFormula1" title=" \int\limits {3 ^{ \frac{x}{2} } } \, dx " alt=" \int\limits {3 ^{ \frac{x}{2} } } \, dx " align="absmiddle" class="latex-formula">=
Mathematics
2 answers:
Kryger [21]3 years ago
7 0
\int 3^{\tfrac{x}{2}}\, dx=(*)\\
t=\dfrac{x}{2}\\
dt=\dfrac{x}{2}\, dx\\
dx=2\, dt\\
(*)=\int 3^t\cdot2\, dt=\\
2\int 3^t \, dt=\\
2\cdot\dfrac{3^t}{\ln 3}+C=\\
\boxed{\dfrac{2\cdot3^{\tfrac{x}{2}}}{\ln 3}+C}

jekas [21]3 years ago
7 0
\int { { 3 }^{ \frac { x }{ 2 }  } } dx\\ \\ =\int { { \left( { 3 }^{ x } \right)  }^{ \frac { 1 }{ 2 }  } } dx

However:

u={ 3 }^{ x }\\ \\ \therefore \quad \frac { du }{ dx } ={ 3 }^{ x }\cdot \ln { 3 } \\ \\ \therefore \quad du={ 3 }^{ x }\cdot \ln { 3 } dx\\ \\ \therefore \quad dx=\frac { 1 }{ { 3 }^{ x }\cdot \ln { 3 }  } du=\frac { 1 }{ u\cdot \ln { 3 }  } du

So let's use:

\int { { u }^{ \frac { 1 }{ 2 }  } } \cdot \frac { 1 }{ u\cdot \ln { 3 }  } du\\ \\ =\int { \frac { 1 }{ \ln { 3 }  }  } \cdot { u }^{ -\frac { 1 }{ 2 }  }du

But you need to know that:

\int { k{ u }^{ n } } du\\ \\ =\frac { k{ u }^{ n+1 } }{ n+1 } +C

Therefore:

\int { \frac { 1 }{ \ln { 3 }  }  } \cdot { u }^{ -\frac { 1 }{ 2 }  }du\\ \\ =\frac { \frac { 1 }{ \ln { 3 }  } \cdot { u }^{ \frac { 1 }{ 2 }  } }{ \frac { 1 }{ 2 }  } +C

\\ \\ =\frac { 1 }{ \ln { 3 }  } \cdot { u }^{ \frac { 1 }{ 2 }  }\cdot 2+C\\ \\ =\frac { 1 }{ \ln { 3 }  } \cdot { 3 }^{ \frac { x }{ 2 }  }\cdot 2+C\\ \\ =\frac { 2\cdot { 3 }^{ \frac { x }{ 2 }  } }{ \ln { 3 }  } +C
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