Question

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

Answer 1

$I=\int a^{mx}{b}^{nx}dx$ ___ (i)

$=\int a^{mx}.b^{nx}dx$

$=a^{mx}\int b^{nx}dx-\int \left(\frac{d}{dx}\right(a^{mx}).\int b^{nx}dx)dx$

$=a^{mx}.\frac{b^{nx}}{nlnb}-\int a^{mx}\times m\times lna\times \frac{b^{nx}}{nlnb}dx$

$I=\frac{a^{mx}{b}^{nx}}{nlnb}-\frac{m}{n}\frac{lna}{lnb}\int a^{mx}{b}^{nx}dx$

$\Rightarrow I+\frac{m}{n}\frac{lna}{lnb}I=\frac{a^{mx}{b}^{nx}}{nlnb}$

$\Rightarrow I\left(\frac{nlnb+mlna}{nlnb}\right)=\frac{a^{mx}{b}^{nx}}{nlnb}$

$\Rightarrow I=\frac{a^{mx}{b}^{nx}}{mlna+nlnb}$