Calculus

Integration of Exponential Functions

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

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

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

What is the value of $x$ that satisfies $\sin \left( \frac{\pi}{2} \log_{x^3}\left( \frac{d}{dx}\left( \int x dx \right)\right)\right)=x^2-8x-\frac{65}{2}?$

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

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