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(C) The matrix $A$ is a diagonal matrix,so its determinant $|A|$ is the product of its diagonal elements: $|A| = a 	imes b 	imes c = 7^x 	imes 7^{7

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$\frac{7^{7^{7^x}}}{(\log 7)^3} + k$,where $k$ is constant of integration

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(C) The matrix $A$ is a diagonal matrix,so its determinant $|A|$ is the product of its diagonal elements: $|A| = a 	imes b 	imes c = 7^x 	imes 7^{7^x} 	imes 7^{7^{7^x}}$. Using the property of exponents $a^m 	imes a^n = a^{m+n}$,we have: $|A| = 7^{x + 7^x + 7^{7^x}}$. We need to evaluate the integral $I = \int 7^{x + 7^x + 7^{7^x}} \, dx$. This integral does not have a standard elementary form. However,checking the derivative of the function $f(x) = 7^{7^{7^x}}$,we use the chain rule: $\frac{d}{dx} (7^{7^{7^x}}) = 7^{7^{7^x}} \cdot \log 7 \cdot \frac{d}{dx} (7^{7^x}) = 7^{7^{7^x}} \cdot \log 7 \cdot 7^{7^x} \cdot \log 7 \cdot \frac{d}{dx} (7^x) = 7^{7^{7^x}} \cdot 7^{7^x} \cdot 7^x \cdot (\log 7)^3$. Therefore,$\frac{d}{dx} \left( \frac{7^{7^{7^x}}}{(\log 7)^3} ight) = 7^{7^{7^x}} \cdot 7^{7^x} \cdot 7^x = |A|$. Thus,$\int |A| \, dx = \frac{7^{7^{7^x}}}{(\log 7)^3} + k$.

If $A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}$ where $a = 7^x$,$b = 7^{7^x}$,$c = 7^{7^{7^x}}$,then $\int |A| \, dx$ (where $|A|$ is the determinant of the matrix $A$) is equal to:

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