Integrate - e x a x dx

Using integration by parts ⇒ I=e^x. ∫ a^x dx+∫ddx (e^x). ∫ a^x dx. dx ⇒ I=e^x a^x log(a) +∫ e^x a^x log(a) ⇒ I=e^x a^x log(a) +log(a) ∫ e^x a^x dx ...

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Byju's Answer

Standard XII

Mathematics

Integration by Substitution

Integrate ∫...

Question

Integrate $\int e^{x} a^{x} d x$

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Solution

Using integration by parts

$\Rightarrow I = e^{x} . \int a^{x} d x + \int \frac{d}{d x} (e^{x}) . \int a^{x} d x . d x$

$\Rightarrow I = e^{x} a^{x} l o g (a) + \int e^{x} a^{x} l o g (a)$

$\Rightarrow I = e^{x} a^{x} l o g (a) + l o g (a) \int e^{x} a^{x} d x$

$\Rightarrow I = e^{x} a^{x} l o g (a) + l o g (a) (I)$

$\Rightarrow I = \frac{e^{x} a^{x} l o g (a)}{1 - l o g (a)}$

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Integrate $\int e^{x} a^{x} d x$