Question

$\int 3^{3^{3^x}}{3}^{3^x}{3}^{x}dx=$

• A. $\frac{3^{3^x}}{{\left(\log 3\right)}^3}+c$
• B. $\frac{3^{3^{3^x}}}{{\left(\log 3\right)}^3}+c$
• C. $3^{3^x}{\left(\log 3\right)}^3+c$
• D. $3^{3^{3^x}}{\left(\log 3\right)}^3+c$

Answer 1

The correct option is D $3^{3^{3^x}}{\left(\log 3\right)}^3+c$

$I=\int 3^{3^{3^x}}{3}^{3^x}{3}^{x}dx$

Let $3^x=t$

$3^{x}dx=\frac{1}{{\log }_{e}3}dt$ [ Differentiating both sides ]

$I=\frac{1}{\log 3}\int 3^{3^t}{.3}^{t}dt$

$I=\frac{1}{(\log 3{)}^2}\int 3^{3^t}{.3}^{t}dt$

Let $3^t=v$

$3^t.\log 3dt=dv$

$3^{t}dt=\frac{dv}{\log 3}$

$I=\frac{3^v}{(\log 3{)}^3}+c$

$I=\frac{3^{3^{3^x}}}{(\log 3{)}^3}+c$