How do you find the antiderivative of #(5x^2)/(x^2 + 1)#?
The integral becomes:
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To find the antiderivative of (\frac{5x^2}{x^2 + 1}), you can use the method of partial fraction decomposition. First, rewrite the fraction as:
[\frac{5x^2}{x^2 + 1} = \frac{Ax + B}{x^2 + 1}]
Then, multiply both sides by (x^2 + 1) to clear the denominator:
[5x^2 = (Ax + B)(x^2 + 1)]
Expand the right side:
[5x^2 = Ax^3 + Ax + Bx^2 + B]
Now, equate coefficients of like terms:
For (x^2) terms: (0x^2 = B) For (x) terms: (5x = Ax) For constant terms: (0 = A)
From the constant term equation, (A = 0), and from the (x) term equation, (5x = Ax) implies (A = 5).
So, the partial fraction decomposition is:
[\frac{5x^2}{x^2 + 1} = \frac{5}{x^2 + 1}]
Now, you can integrate term by term. The integral of (\frac{5}{x^2 + 1}) is (5\tan^{-1}(x) + C), where (C) is the constant of integration.
Therefore, the antiderivative of (\frac{5x^2}{x^2 + 1}) is (5\tan^{-1}(x) + C).
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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.
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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