What is the value of \alpha ?
Consider the following for the next two (02) items that follow : Let \int\frac{dx}{\sqrt{x+1}-\sqrt{x-1}}=\alpha(x+1)^{\frac{3}{2}}+\beta(x-1)^{\frac{3}{2}}+c
- A. \frac{1}{3} ✓
- B. \frac{2}{3}
- C. 1
- D. \frac{4}{3}
Correct Answer: A. \frac{1}{3}
Explanation
Rationalize the denominator by multiplying the numerator and denominator by \sqrt{x+1}+\sqrt{x-1}. The integral becomes \int \frac{\sqrt{x+1}+\sqrt{x-1}}{(x+1)-(x-1)}\,dx = \frac{1}{2}\int ( (x+1)^{1/2} + (x-1)^{1/2} )\,dx. Evaluating this gives \frac{1}{2} \left[ \frac{2}{3}(x+1)^{3/2} + \frac{2}{3}(x-1)^{3/2} \right] + c = \frac{1}{3}(x+1)^{3/2} + \frac{1}{3}(x-1)^{3/2} + c. Comparing this with the given expression, \alpha = \frac{1}{3}.
Related questions on Calculus
- Let z=[y] and y=[x]-x, where [.] is the greatest integer function. If x is <strong>NOT</strong> an integer but positive, then what i...
- If f(x)=4x+1 and g(x)=kx+2 such that fog(x)=gof(x), then what is the value of k?
- What is the <strong>MINIMUM</strong> value of the function f(x)=\log_{10}(x^{2}+2x+11)?
- What is \int(x^{x})^{2}(1+\ln x)\,dx equal to ?
- What is \int e^{x}\{1+\ln x+x\ln x\}\,dx equal to?