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Integration of Exponential Functions


Let N=074e4x dxN = \displaystyle \int_0^{7} 4 e^{4 x}\ dx. If N=eabN = e^{a} - b, where aa and bb are positive integers, what is the value of a+ba + b?

If f(x)f(x) is a function such that f(x)=ex+2dxf(x)=\int e^{x+2} dx and f(2)=6,f(-2)=6, what is the value of f(x)?f(x)?

If f(0)=7154ln4\displaystyle f(0)=7-\frac{15}{4 \ln 4} for the function f(x)=(2x+2x)2dx,f(x)=\int \left( 2 ^x + 2 ^{-x} \right)^2 dx, what is the value of f(1)?f(1)?

What is the value of xx that satisfies sin(π2logx3(ddx(xdx)))=x28x652?\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}?

If f(x)=27x643x4dxf(x)=\int \frac{27 ^x-64}{3 ^x-4} dx and f(0)=4ln3,\displaystyle f(0)=\frac{4}{\ln 3}, what is the value of f(1)?f(1)?


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