Mathematics

$$\int {{3^x}{3^{{3^x}}}{3^{{3^{{3^x}}}}}} dx$$ is equal to 


ANSWER

$$\dfrac{{{3^{{3^{{3^x}}}}}}}{{{{\left( {\log 3} \right)}^3}}} + C$$


SOLUTION
$$I\displaystyle\int 3^x3^{3^x}3^{3^{3^x}}dx$$

put $$3^x=p$$

$$3^xlog 3dx=dp$$
$$\Rightarrow I=\dfrac{1}{log 3}\displaystyle\int 3^p3^{3^p}dp$$

put $$3^p=k$$

$$3^plog 3dp=dk$$
$$I=\dfrac{1}{(log 3)^2}\displaystyle\int 3^kdk$$
$$\Rightarrow I=\dfrac{1}{(log 3)^3}3^k$$

$$=\dfrac{1}{(log 3)^3}3^{3^p}=\dfrac{1}{(log 3)^3}3^{3^{3^x}}$$

$$\therefore \displaystyle\int 3^x3^{3^x}3^{3^{3^x}}dx=\dfrac{1}{(log3)^3}3^{3^{3^x}}+c$$.
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Single Correct Medium Published on 17th 09, 2020
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