1 log a ( a x cos a x ) ? d x =

Question

$\int \frac{1}{\log a} (a^{x} \cos a^{x}) d x =$

Smart Achievers · Accepted Answer

(3)

Ans. $\frac{1}{{(\log a)}^{2}} \sin a^{x} + C$

Explanation:

Put $a^{x} = t$

Then, $a^{x} \log a d x = d t$

$\Rightarrow$ $a^{x} d x = \frac{d t}{\log a}$

Therefore, $\int \frac{1}{\log a} (a^{x} \cos a^{x}) d x = \int \frac{1}{{(\log a)}^{2}} \cos t d t$

$= \frac{1}{{(\log a)}^{2}} \sin t + C$

$= \frac{1}{{(\log a)}^{2}} \sin a^{x} + C$

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