Antiderivative Question for ∫a^x dx: Is it a^x/ln a or a^x/ln|a|?

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In summary, an antiderivative is the inverse of taking a derivative and is a function whose derivative is equal to the original function. It can be found using integration techniques such as u-substitution, integration by parts, or the power rule. Unlike a definite integral, an antiderivative is not a numerical value and does not have specific limits of integration. However, not all functions have antiderivatives, as they must be continuous and have a well-defined derivative. Antiderivatives are useful in mathematics and science for solving problems involving rates of change, finding areas under curves, and solving optimization problems.
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tmt1
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for ∫a^x dx,

is the answer a^x/ln a or a^x/ln|a| or does it matter?

Thanks
 
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  • #2
The function $a^x$ is not defined for negative values of $a$. If $a>0$ and $a\not = 1$ then what you wrote is correct without absolute values.
 

FAQ: Antiderivative Question for ∫a^x dx: Is it a^x/ln a or a^x/ln|a|?

What is an antiderivative?

An antiderivative, also known as the indefinite integral, is the reverse process of taking a derivative. It is a function whose derivative is equal to the original function.

How do you find the antiderivative of a function?

To find the antiderivative of a function, you can use integration techniques such as u-substitution, integration by parts, or the power rule. It is important to note that the antiderivative of a function is not unique, as there can be an infinite number of functions with the same derivative.

What is the difference between an antiderivative and a definite integral?

An antiderivative is the opposite of taking a derivative and is a function, while a definite integral is the numerical value of the area under a curve. A definite integral has specific limits of integration, while an antiderivative does not.

Can all functions have an antiderivative?

No, not all functions have an antiderivative. A function must be continuous and have a well-defined derivative in order to have an antiderivative. Functions that have discontinuities or vertical asymptotes do not have antiderivatives.

How is an antiderivative useful in mathematics and science?

An antiderivative is useful in mathematics and science for solving problems involving rates of change, such as velocity and acceleration in physics, or growth and decay in biology. It is also used in finding areas under curves, calculating work and displacement, and solving optimization problems.

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