Compute the indefinite integral ∫ (2x)/(x^2+1) dx.

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Multiple Choice

Compute the indefinite integral ∫ (2x)/(x^2+1) dx.

Explanation:
The integrand fits the pattern f′(x)/f(x), whose antiderivative is ln|f(x)| + C. Here f(x) = x^2 + 1 and f′(x) = 2x, so ∫ (2x)/(x^2+1) dx = ln|x^2+1| + C. Since x^2+1 > 0 for all x, this is ln(x^2+1) + C. Differentiating confirms the result: d/dx[ln(x^2+1)] = (2x)/(x^2+1). Other options don’t match the derivative: arctan x would give 1/(1+x^2), (1/2)ln(x^2+1) differentiates to x/(x^2+1), and ln(x+1) gives 1/(x+1).

The integrand fits the pattern f′(x)/f(x), whose antiderivative is ln|f(x)| + C. Here f(x) = x^2 + 1 and f′(x) = 2x, so ∫ (2x)/(x^2+1) dx = ln|x^2+1| + C. Since x^2+1 > 0 for all x, this is ln(x^2+1) + C. Differentiating confirms the result: d/dx[ln(x^2+1)] = (2x)/(x^2+1). Other options don’t match the derivative: arctan x would give 1/(1+x^2), (1/2)ln(x^2+1) differentiates to x/(x^2+1), and ln(x+1) gives 1/(x+1).

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