Mathematics

$$\displaystyle\int a^{mx}b^{nx}\ dx$$


SOLUTION
$$\int { { a }^{ mx } } { b }^{ nx }dx$$
$$\Rightarrow \quad Putting\quad { a }^{ mx }{ b }^{ nx }=t$$
$$dx\left[ \left( { a }^{ mx }loga \right) { b }^{ nx }+\left( { b }^{ nx }logb \right) { a }^{ nx } \right] =dt$$
or    $$dx=\dfrac { dt }{ t\left[ loga+logb \right]  } $$
$$\Rightarrow \int { \dfrac { tdt }{ t\left[ loga+logb \right]  }  } $$
$$=\dfrac { t }{ \left[ loga+logb \right]  } +c$$
$$=\dfrac { { a }^{ mx }{ b }^{ nx } }{ \left[ loga+logb \right]  } +c$$
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Subjective Hard Published on 17th 09, 2020
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