What is |U^{2}(x)-V^{2}(x)| equal to?
Direction: Consider the following for the two (02) items that follow : Let 2\int\frac{x^{2}-1}{\sqrt{x^{2}+1}}\,dx=U(x)V(x)-3\ln\{U(x)+V(x)\}+c
- A. 0
- B. 1 ✓
- C. 2
- D. 3
Correct Answer: B. 1
Explanation
We can rewrite the integral as \int \frac{x^2+1-2}{\sqrt{x^2+1}}\,dx = \int \sqrt{x^2+1}\,dx - 2\int \frac{1}{\sqrt{x^2+1}}\,dx. Evaluating gives \left[\frac{x}{2}\sqrt{x^2+1} + \frac{1}{2}\ln|x+\sqrt{x^2+1}|\right] - 2\ln|x+\sqrt{x^2+1}| = \frac{x}{2}\sqrt{x^2+1} - \frac{3}{2}\ln|x+\sqrt{x^2+1}|. Multiplying the result by 2 gives x\sqrt{x^2+1} - 3\ln|x+\sqrt{x^2+1}|. Comparing this to the given expression, U(x) = x and V(x) = \sqrt{x^2+1} (or vice versa). Thus, |U^2(x) - V^2(x)| = |x^2 - (x^2+1)| = |-1| = 1.
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