# What are examples of functions that cannot be integrated?

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It depends on what you mean. Do you mean you can't find a formula for an antiderivative? Or do you mean the definite integral doesn't exist?

This last function is, however, "Lebesgue integrable" (pronounced "Lah-bagh" with a long "a" sound in the second syllable). I won't get into details, but in a nutshell, there are lots of "theories of integration" with respect to which a given function might be integrable or not.

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Examples of functions that cannot be integrated include:

- ( e^{-x^2} ) (Gaussian function)
- ( \frac{\sin(x)}{x} ) (Sine integral function)
- ( \frac{1}{x} ) (Reciprocal function)
- ( \sqrt{1 + x^2} ) (Square root function with non-polynomial argument)
- ( \frac{\sin(x)}{x^2} ) (Sine integral function multiplied by a power of ( x ))
- ( \ln(x) ) (Natural logarithm function)
- ( \frac{1}{x \ln(x)} ) (Logarithmic integral function)

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the sum of the infinite series #Sigma(1/10)^k# from k=1 to #oo#?
- What is #int ln4x^2dx#?
- What is the antiderivative of #e^(-3x)#?
- How do you use sigma notation to write the sum for #1/(1*3)+1/(2*4)+1/(3*5)+...+1/(10*12)#?
- How do you integrate #(5x)/(2x^2+11x+12)# using partial fractions?

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