What is \int_{1}^{3}f(x)\,dx equal to?
Direction: Consider the following for the two (02) items that follow : Let f(x)=|x^{2}-x-2|.
- A. 2
- B. 3 ✓
- C. 4
- D. 5
Correct Answer: B. 3
Explanation
We split the integral at x=2 because the sign of x^2-x-2 changes there. \int_1^3 |x^2-x-2|\,dx = \int_1^2 (2+x-x^2)\,dx + \int_2^3 (x^2-x-2)\,dx. The first part is [2x + \frac{x^2}{2} - \frac{x^3}{3}]_1^2 = \frac{10}{3} - (2 + \frac{1}{2} - \frac{1}{3}) = \frac{10}{3} - \frac{13}{6} = \frac{7}{6}. The second part is [\frac{x^3}{3} - \frac{x^2}{2} - 2x]_2^3 = (9 - \frac{9}{2} - 6) - (\frac{8}{3} - 2 - 4) = -\frac{3}{2} + \frac{10}{3} = \frac{11}{6}. The sum is \frac{7}{6} + \frac{11}{6} = \frac{18}{6} = 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?