# How do you find the antiderivative of #int 1/(x^2+10x+21) dx#?

So:

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To find the antiderivative of ( \int \frac{1}{x^2 + 10x + 21} , dx ), we first need to factor the denominator. The expression ( x^2 + 10x + 21 ) factors into ( (x + 7)(x + 3) ).

Therefore, the integral becomes ( \int \frac{1}{(x + 7)(x + 3)} , dx ).

Next, we use partial fraction decomposition to break down the fraction into simpler fractions. We express ( \frac{1}{(x + 7)(x + 3)} ) as ( \frac{A}{x + 7} + \frac{B}{x + 3} ).

After finding the values of ( A ) and ( B ), we integrate each term separately.

Once integrated, the antiderivative of ( \frac{1}{x^2 + 10x + 21} ) will be expressed as a sum of the antiderivatives of the simpler fractions, each corresponding to the terms ( \frac{A}{x + 7} ) and ( \frac{B}{x + 3} ), plus a constant of integration.

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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.

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