int_{-4}^{4} (2^x + 2^{-x})(3^x + 3^{-x}) , d | vedclass

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(B) Let $f(x) = (2^x + 2^{-x})(3^x + 3^{-x})$.Expanding the product,we get $f(x) = 2^x 3^x + 2^x 3^{-x} + 2^{-x} 3^x + 2^{-x} 3^{-x} = 6^x + (2/3)^x +

Vedclass · Accepted Answer

$4 \int_{0}^{4} (6^x + 6^{-x}) \, dx$

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(B) Let $f(x) = (2^x + 2^{-x})(3^x + 3^{-x})$.Expanding the product,we get $f(x) = 2^x 3^x + 2^x 3^{-x} + 2^{-x} 3^x + 2^{-x} 3^{-x} = 6^x + (2/3)^x + (3/2)^x + 6^{-x}$.Since $f(-x) = 6^{-x} + (2/3)^{-x} + (3/2)^{-x} + 6^x = 6^{-x} + (3/2)^x + (2/3)^x + 6^x = f(x)$,the function $f(x)$ is an even function.Using the property $\int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx$ for even functions,we have:$\int_{-4}^{4} f(x) \, dx = 2 \int_{0}^{4} (6^x + (2/3)^x + (3/2)^x + 6^{-x}) \, dx$.This can be written as $2 \int_{0}^{4} (6^x + 6^{-x} + (2/3)^x + (3/2)^x) \, dx$.

$\int_{-4}^{4} (2^x + 2^{-x})(3^x + 3^{-x}) \, dx$ is equal to

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