$\int \frac{5^{x}}{\sqrt{5(-2 x)-5^{(2 x)}}} d x=$

Question

Aicharya · Accepted Answer

Let $\begin{aligned} I &=\int \frac{5^{x}}{\sqrt{\left(5^{2 x}ight)^{-1}-5^{2 x}}} d x \ &=\int \frac{5^{x}}{\sqrt{\left(\frac{1}{5^{2 x}}ight)-5^{2 x}}} d x=\int \frac{5^{x} \cdot 5^{x}}{\sqrt{1-\left(5^{2 x}ight)}} d x \end{aligned}$ Put $5^{2 \mathrm{x}}=\mathrm{t} \Rightarrow(2 \log 5) 5^{2 \mathrm{x}} \mathrm{dx}=\mathrm{dt} \Rightarrow 5^{\mathrm{x}} \cdot 5^{\mathrm{x}} \mathrm{dx}=\frac{\mathrm{dt}}{\log 25}$ $\begin{aligned} 	herefore \mathrm{I} &=\int \frac{1}{\sqrt{1-\mathrm{t}^{2}}} 	imes \frac{\mathrm{dt}}{(\log 25)}=\frac{1}{\log 25} \int \frac{\mathrm{dt}}{\sqrt{1-\mathrm{t}^{2}}} \mathrm{dt} \ &=\frac{1}{\log 25} \sin ^{-1} \mathrm{t}+\mathrm{c}=\frac{1}{\log 25} \sin ^{-1}\left(5^{2 \mathrm{x}}ight)+\mathrm{c} \end{aligned}$

$\int \frac{5^{x}}{\sqrt{5(-2 x)-5^{(2 x)}}} d x=$

Solution