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Let N=∫074e4x dxN = \displaystyle \int_0^{7} 4 e^{4 x}\ dxN=∫074e4x dx. If N=ea−bN = e^{a} - bN=ea−b, where aaa and bbb are positive integers, what is the value of a+ba + ba+b?
If f(x)f(x)f(x) is a function such that f(x)=∫ex+2dxf(x)=\int e^{x+2} dxf(x)=∫ex+2dx and f(−2)=6,f(-2)=6,f(−2)=6, what is the value of f(x)?f(x)?f(x)?
If f(0)=7−154ln4\displaystyle f(0)=7-\frac{15}{4 \ln 4}f(0)=7−4ln415 for the function f(x)=∫(2x+2−x)2dx,f(x)=\int \left( 2 ^x + 2 ^{-x} \right)^2 dx,f(x)=∫(2x+2−x)2dx, what is the value of f(1)?f(1)?f(1)?
What is the value of xxx that satisfies sin(π2logx3(ddx(∫xdx)))=x2−8x−652?\sin \left( \frac{\pi}{2} \log_{x^3}\left( \frac{d}{dx}\left( \int x dx \right)\right)\right)=x^2-8x-\frac{65}{2}?sin(2πlogx3(dxd(∫xdx)))=x2−8x−265?
If f(x)=∫27x−643x−4dxf(x)=\int \frac{27 ^x-64}{3 ^x-4} dxf(x)=∫3x−427x−64dx and f(0)=4ln3,\displaystyle f(0)=\frac{4}{\ln 3},f(0)=ln34, what is the value of f(1)?f(1)?f(1)?
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