Evaluate the integral by substitution: integral of [2x / (x^2 + 9)] dx.
Answer and explanation
Correct answer: C. ln(x^2 + 9) + C
- Step 1: Let u = x^2 + 9, so that du = 2x dx.
- Step 2: Substitute into the integral: integral of (1 / u) du.
- Step 3: Evaluate the natural logarithm integral: ln|u| + C = ln(x^2 + 9) + C (since x^2 + 9 > 0).
Answer
ln(x^2 + 9) + C.
Why not the others: '2 * ln(x^2 + 9) + C' doubles the coefficient instead of absorbing 2x into du; '1 / (x^2 + 9) + C' treats the fraction as a power rule case with exponent -1; 'arctan(x/3) + C' confuses the logarithmic form with the standard arctangent integral which lacks x in the numerator.
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