Evaluate exactly: $\int_{-5 x}^{5 x} e^{2 x} \cos 2 x d x$ .
A) $\frac{1}{4}\left(e^{10 x}-e^{-10 x}\right)$
B) $\frac{1}{4}\left(e^{5 x}-e^{-5 x}\right)$
C) $\frac{1}{4}\left(e^{10 x}+e^{-10 x}\right)$
D) $\frac{1}{4} e^{10 x}$

Question

Quizplus · Accepted Answer

The integral of $e^{2x} \cos 2x dx$ over the symmetric interval $[-5x, 5x]$ results in a function involving hyperbolic sine or cosine due to the properties of even and odd functions. However, since the integral's limits are symmetric and involve an exponential function multiplied by a cosine function, the result should reflect both the growth and decay characteristics of the exponential function. Option A, $\frac{1}{4}\left(e^{10 x}-e^{-10 x}\right)$, correctly represents this as it is the only option that includes both $e^{10x}$ and $e^{-10x}$ in a subtractive form, which is characteristic of the hyperbolic sine function, a common result in integrals of this type.

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Evaluate Exactly $\int_{-5 x}^{5 x} e^{2 x} \cos 2 x d x$