MathDB
Today's calculation of Integral 833

Source: 2012 Kumamoto University entrance exam/medicine

July 4, 2012
calculusintegrationtrigonometryderivativecomplex numberscalculus computations

Problem Statement

Let f(x)=∫0xet(cos⁡t+sin⁡t) dt, g(x)=∫0xet(cos⁡t−sin⁡t) dt.f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.
For a real number aa, find ∑n=1∞e2a{f(n)(a)}2+{g(n)(a)}2.\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.
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