Question

$\int 5^{5^{5^x}}{.5}^{5^x}{.5}^{x}dx$ is equal to

• A. $\frac{5^{5^x}}{{\left(\ell n5\right)}^3}+c$
• 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

Answer: (C)

$\frac{5^{5^{5^x}}}{{\left(\ell n5\right)}^3}+c$

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

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

Let's differentiate

$5^{5^{5^x}}{.5}^{5^x}{.5}^x{\left(\ell n5\right)}^{3}dx=dt$

$I=\frac{1}{{\left(\ell n5\right)}^3}\int dt=\frac{t}{{\left(\ell n5\right)}^3}+c$

$=\frac{5^{5^{5^x}}}{{\left(\ell n5\right)}^3}+c$