What is the antiderivative of the square root of the quantity (1+x^2)
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To find the antiderivative of the square root of the quantity (1+x^2), we can use the substitution method.
Let's make the substitution u = 1 + x^2. Then, differentiating both sides with respect to x, we get du/dx = 2x.
Rearranging, we have dx = du/(2x).
Now, substitute u and dx into the original integral:
∫ √(1+x^2) dx = ∫ √u * (du/(2x))
Since dx = du/(2x),
= (1/2) ∫ (√u/u) du
Now, we can simplify the integral by splitting the fraction inside:
= (1/2) ∫ u^(-1/2) du
Integrating u^(-1/2) with respect to u gives us 2 * u^(1/2):
= (1/2) * 2 * u^(1/2) + C
= u^(1/2) + C
Replacing u with the original substitution, we have the final answer:
= (1 + x^2)^(1/2) + C
So, the antiderivative of the square root of the quantity (1+x^2) is (1 + x^2)^(1/2) + C.
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