Question

Solve $\int 5^{5^{5^x}}{.5}^{5^x}{.5}^{x}dx$

• A. $\frac{5^{5^{5^x}}}{(ln5{)}^3}+c$
• B. $5^{5^{5^x}}{\left(\ell n5\right)}^3+c$
• C. $\frac{5^{5^{5^x}}}{{\left(\ell n5\right)}^3}+c$
• D. none of these

Answer 1

The correct option is A $\frac{5^{5^{5^x}}}{(ln5{)}^3}+c$

Let $I=\int 5^{5^{5^x}}{.5}^{5^x}{.5}^{x}dx$

Put $5^{5^{5^x}}=t$

$\int 5^{5^{5^x}}{.5}^{5^x}{.5}^x(\ln 5{)}^{3}dx=dt$ $(Since{a}^x=a^x\ln a)$

So,

$I=\frac{1}{(ln5{)}^3}\int dt=\frac{t}{(ln5{)}^3}+c$

$=\frac{5^{5^{5^x}}}{(ln5{)}^3}+c$