a-mx-b-nx-dx-

Question Number 152110 by peter frank last updated on 25/Aug/21

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

Answered by peter frank last updated on 25/Aug/21

\int {(a^{m} b^{n})}^{x}

\dots \dots

Answered by Olaf_Thorendsen last updated on 26/Aug/21

F (x) = \int a^{m x} b^{n x} d x

F (x) = \int e^{(m \ln a + n \ln b) x} d x

F (x) = \frac{e^{(m \ln a + n \ln b) x}}{m \ln a + n \ln b} + C

F (x) = \frac{a^{m x} b^{n x}}{\ln (a^{m} b^{n})} + C

Commented by peter frank last updated on 26/Aug/21

thank you

Answered by puissant last updated on 26/Aug/21

K = \int a^{m x} b^{n x} d x

= a^{m x} \int b^{n x} d x - \int (\frac{d}{d x} (a^{m x}) . \int b^{n x} d x)

= a^{m x} \frac{b^{n x}}{n l n b} - \int a^{m x} \times m l n a \frac{b^{n x}}{n l n b} d x

= \frac{a^{m x} b^{n x}}{n l n b} - \frac{m l n a}{n l n b} \int a^{m x} b^{n x} d x

\Rightarrow K = \frac{m l n a}{n l n b} \times K = \frac{a^{m x} b^{n x}}{n l n b}

\Rightarrow K (\frac{n l n b + m l n a}{n l n b}) = \frac{a^{m x} b^{n x}}{n l n b}

\Rightarrow ∴∵ K = \frac{a^{m x} b^{n x}}{m l n a + n l n b} + C . .

Commented by peter frank last updated on 26/Aug/21

thank you