Definite Integration
Indefinite Integration
Grade Class 12
Question:
∫ (x^2 - 1) / (x^3 * sqrt(2x^4 - 2x^2 + 1)) dx is equal to -
(A) (sqrt(2x^4 - 2x^2 + 1)) / x^2 + c
(B) (sqrt(2x^4 - 2x^2 + 1)) / x^3 + c
(C) (sqrt(2x^4 - 2x^2 + 1)) / x + c
(D) (sqrt(2x^4 - 2x^2 + 1)) / 2x^2 + c
Step-by-Step Solution
Key Concept: Divide the numerator and denominator by x^3 to simplify the integral into a form suitable for substitution.
The integral is \int (x^2 - 1) / (x^3 * sqrt(2x^4 - 2x^2 + 1)) dx. Divide numerator and denominator by x^3: \int (1/x - 1/x^3) / sqrt(2x - 2/x + 1/x^3) dx. Let t = 2x - 2/x + 1/x^3. Then dt = (2 + 2/x^2 - 3/x^4) dx. This approach is complex. Alternatively, rewrite as \int (1/x^2 - 1/x^4) / sqrt(2 - 2/x^2 + 1/x^4) dx. Let u = 2 - 2/x^2 + 1/x^4. Then du = (4/x^3 - 4/x^5) dx = 4/x^3 * (1 - 1/x^2) dx. The integral becomes (1/4) \int du / sqrt(u) = (1/4) * 2 * sqrt(u) + C = (1/2) * sqrt(2 - 2/x^2 + 1/x^4) + C = (1/2) * sqrt((2x^4 - 2x^2 + 1) / x^4) + C = sqrt(2x^4 - 2x^2 + 1) / (2x^2) + C.
Correct Answer: D