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Antiderivatives

This is the opposite of the derivative - and it's an integral part of Calculus. It's uses range from basic integrals to differential equations, with applications in Physics, Chemistry, and Economics.

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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