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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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