Mathematics

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


SOLUTION
$$I=\int a^{mx}b^{nx}dx$$ ___ (i)
$$=\int a^{mx}.b^{nx}dx$$
$$=a^{mx}\int b^{nx}dx-\int (\frac{d}{dx}(a^{mx}).\int b^{nx}dx)dx$$
$$=a^{mx}.\frac{b^{nx}}{nlnb}-\int a^{mx}\times m\times lna\times \frac{b^{nx}}{nlnb}dx$$
$$I=\frac{a^{mx}b^{nx}}{nlnb}-\frac{m}{n}\frac{lna}{lnb}\int a^{mx}b^{nx}dx$$
$$\Rightarrow I+\frac{m}{n}\frac{lna}{lnb}I=\frac{a^{mx}b^{nx}}{nlnb}$$
$$\Rightarrow I(\frac{nlnb+mlna}{nlnb})=\frac{a^{mx}b^{nx}}{nlnb}$$
$$\Rightarrow I=\frac{a^{mx}b^{nx}}{mlna+nlnb}$$
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Subjective Medium Published on 17th 09, 2020
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