Problems & Solutions
Find the antiderivative of each function.
\(\textbf{1)}\) \(\displaystyle f(x)= x+5 \)
\(\textbf{2)}\) \(\displaystyle f(x)= x^4-2x^2+5x \)
\(\textbf{3)}\) \(\displaystyle f(x)= x^{1.5}+5x \)
\(\textbf{4)}\) \(\displaystyle f(x)= \displaystyle \frac{1}{x^5} \)
\(\textbf{5)}\) \(\displaystyle f(x)= \sqrt[3]{x}+2 \)
\(\textbf{6)}\) \(\displaystyle f(x)= \displaystyle \frac{x^3+5x^2-4}{\sqrt{x}} \)
\(\textbf{7)}\) \(\displaystyle f(x)= x^3{\sqrt{x}} \)
\(\textbf{8)}\) \(\displaystyle f(x)= 0 \)
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In Summary
An antiderivative, also known as an indefinite integral, is a mathematical concept used to find the original function given its derivative. It is the reverse process of taking a derivative.
Antiderivatives are related to integrals and are an important in mathematics because they allow us to solve problems involving the accumulation of quantities, such as distance. They are used in fields such as physics, engineering, and economics to model real-world phenomena.
Antiderivatives are typically introduced in a calculus course. They are a fundamental concept in calculus, and are used throughout the subject.
Common mistakes made when working with antiderivatives include forgetting to add the constant (+C).
Real world examples of antiderivatives
Distance traveled: The antiderivative of speed (rate of change of distance with respect to time) is distance traveled.
Work done: The antiderivative of force (rate of change of work with respect to distance) is work done.
Electric charge: The antiderivative of electric current (rate of change of electric charge with respect to time) is electric charge.