Question

If \( \int(x+2) \sqrt{x^{2}-x+2} d x=\frac{1}{3} f(x)+\frac{5}{8} g(x)+\frac{35}{16} h(x)+c \) then \( f(-1)+g(-1)+h\left(\frac{1}{2}\right)= \)

• A. -4
• B. 2
• C. 4
• D. -2

Hint:

Match coefficients carefully. The term multiplying \( g(x) \) is \( \frac{5}{2} \times \frac{1}{4} = \frac{5}{8} \), which matches the given expression perfectly.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to evaluate the integral by splitting the linear term \( x+2 \) into a part proportional to the derivative of the quadratic and a constant part. Then compare the resulting terms with the given expression to identify functions \( f, g, h \).
Step 2: Key Formula or Approach:
1. Write \( x+2 = A \frac{d}{dx}(x^2-x+2) + B \). 2. Use \( \int \sqrt{Q} Q' dx = \frac{2}{3} Q^{3/2} \). 3. Use \( \int \sqrt{x^2+a^2} dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\sinh^{-1}(\frac{x}{a}) \).
Step 3: Detailed Explanation:
Derivative of \( x^2-x+2 \) is \( 2x-1 \). Express \( x+2 = \frac{1}{2}(2x-1) + \frac{5}{2} \). Integral \( I = \frac{1}{2} \int (2x-1)\sqrt{x^2-x+2} dx + \frac{5}{2} \int \sqrt{x^2-x+2} dx \). Part 1: Let \( u = x^2-x+2 \). \( \frac{1}{2} \int u^{1/2} du = \frac{1}{2} \cdot \frac{2}{3} u^{3/2} = \frac{1}{3} (x^2-x+2)^{3/2} \). Comparing with \( \frac{1}{3}f(x) \implies f(x) = (x^2-x+2)^{3/2} \). Part 2: \( \sqrt{x^2-x+2} = \sqrt{(x-1/2)^2 + 7/4} \). \( \int \sqrt{X^2+a^2} dx = \frac{X}{2}\sqrt{X^2+a^2} + \frac{a^2}{2} \sinh^{-1}(X/a) \). Here \( X = x-1/2 = \frac{2x-1}{2} \), \( a = \frac{\sqrt{7}}{2} \). Integral = \( \frac{5}{2} \left[ \frac{2x-1}{4} \sqrt{x^2-x+2} + \frac{7/4}{2} \sinh^{-1}\left(\frac{2x-1}{\sqrt{7}}\right) \right] \) \( = \frac{5}{8}(2x-1)\sqrt{x^2-x+2} + \frac{35}{16} \sinh^{-1}\left(\frac{2x-1}{\sqrt{7}}\right) \). Comparing: \( g(x) = (2x-1)\sqrt{x^2-x+2} \). \( h(x) = \sinh^{-1}\left(\frac{2x-1}{\sqrt{7}}\right) \). Evaluate: \( f(-1) = ((-1)^2 - (-1) + 2)^{3/2} = (4)^{3/2} = 8 \). \( g(-1) = (2(-1)-1)\sqrt{4} = -3(2) = -6 \). \( h(1/2) = \sinh^{-1}\left(\frac{1-1}{\sqrt{7}}\right) = 0 \). Sum \( = 8 - 6 + 0 = 2 \).
Step 4: Final Answer:
The sum is 2.