int(5^(-x)dx)/(sqrt(5^(2)x+3*5^(x)+1))

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Simplify Integral Expression: Let's start by simplifying the integral expression: int(5^(-x)dx)/(sqrt(5^(2x)+3*5^(x)+1)) We can rewrite 5^(2x) as (5^x)^2 to make the expression inside the square root look like a quadratic in terms of 5^x.Set u = 5^x: Now, let's set u = 5^x. Then du/dx = ln(5) * 5^x. We need to express dx in terms of du, so we solve for dx: dx = du / (ln(5) * 5^x).Substitute u and dx: Substitute u = 5^x and dx = du / (ln(5) * 5^x) into the integral: int(5^(-x)dx)/(sqrt(5^(2x)+3*5^(x)+1)) = int(5^(-x) * du / (ln(5) * 5^x))/(sqrt(u^2+3u+1)) Simplify the integral by canceling out 5^x in the numerator and denominator: = int(du / (ln(5) * sqrt(u^2+3u+1)))Simplify Integral in terms of u: Now we have an integral in terms of u: int(du / (ln(5) * sqrt(u^2+3u+1))) This integral does not have an elementary antiderivative, so we need to use numerical methods or special functions to evaluate it. However, since we are asked for an indefinite integral, we can leave the answer in terms of an integral expression involving u.Express Indefinite Integral: We express the indefinite integral in terms of u and then substitute back to get the final answer in terms of x: Indefinite integral: (1/ln(5)) * int(du / sqrt(u^2+3u+1)) + C Substitute u = 5^x back into the expression: Indefinite integral: (1/ln(5)) * int(du / sqrt((5^x)^2+3*5^x+1)) + CSubstitute back for Final Answer: The final answer in terms of x is: Indefinite integral: (1/ln(5)) * int(du / sqrt((5^x)^2+3*5^x+1)) + C

$\int \frac{5^{-x} d x}{\sqrt{5^{2} x+3 \cdot 5^{x}+1}}$

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∫5−xdx52x+3⋅5x+1 \int \frac{5^{-x} d x}{\sqrt{5^{2} x+3 \cdot 5^{x}+1}} ∫52x+3⋅5x+1​5−xdx​

$\int \frac{5^{-x} d x}{\sqrt{5^{2} x+3 \cdot 5^{x}+1}}$