Definite Integration
Substitution Method
Grade Class 12
Question:
Consider f(x) = \frac{x^2}{1+x^3}; g(t) = \int f(t)dt. If g(1) = 0 then g(x) equals -
(A) \frac{1}{3}ln(1+x^3)
(B) \frac{1}{3}ln\left(\frac{1+x^3}{2}\right)
(C) \frac{1}{2}ln\left(\frac{1+x^3}{3}\right)
(D) \frac{1}{3}ln\left(\frac{1+x^3}{3}\right)
Step-by-Step Solution
Key Concept: Integrate f(x) using substitution u = 1+x^3, then use the condition g(1)=0 to find the constant of integration.
g(x) = integral x^2/(1+x^3) dx = (1/3)ln|1+x^3| + C. Given g(1) = 0, (1/3)ln(2) + C = 0, so C = -(1/3)ln(2). Thus g(x) = (1/3)ln(1+x^3) - (1/3)ln(2) = (1/3)ln((1+x^3)/2).
Correct Answer: B