Anti-Derivative Proof – 27/02/15

y=\frac { 1 }{ \left( n+1 \right)  } { x }^{ \left( n+1 \right)  }\\ \\ \ln { y } =\ln { \left( \frac { { x }^{ \left( n+1 \right)  } }{ \left( n+1 \right)  }  \right)  } \\ \\ \ln { y } =\ln { \left( { x }^{ \left( n+1 \right)  } \right)  } -\ln { \left( \left( n+1 \right)  \right)  } \\ \\ \ln { y } =\ln { \left( { x }^{ \left( n+1 \right)  } \right)  } +{ C }_{ 2 }\\ \\ \therefore \quad { C }_{ 2 }=-\ln { \left( \left( n+1 \right)  \right)  } \\ \\ \ln { y } =\left( n+1 \right) \cdot \ln { x } +{ C }_{ 2 }\\ \\ \frac { 1 }{ y } \cdot \frac { dy }{ dx } =\frac { \left( n+1 \right)  }{ x } \\ \\ y\cdot \frac { 1 }{ y } \cdot \frac { dy }{ dx } =\frac { \left( n+1 \right)  }{ x } \cdot y\\ \\ \frac { dy }{ dx } =\left( n+1 \right) \cdot { x }^{ -1 }\cdot \frac { 1 }{ \left( n+1 \right)  } { x }^{ \left( n+1 \right)  }\\ \\ \frac { dy }{ dx } ={ x }^{ -1 }\cdot { x }^{ \left( n+1 \right)  }\\ \\ \frac { dy }{ dx } ={ x }^{ -1+\left( n+1 \right)  }\\ \\ \frac { dy }{ dx } ={ x }^{ n }\\ \\ \therefore \quad \int { { x }^{ n } } dx=\frac { 1 }{ \left( n+1 \right)  } \cdot { x }^{ n+1 }+C

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