Question

\( \int \sqrt{x^2+x+1} \ dx \)

• A. \( \frac{(2x+1)}{4}\sqrt{x^2+x+1} + \frac{3}{8}\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c \)
• B. \( \frac{x+1}{4}\sqrt{x^2+x+1} + \frac{3}{8}\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c \)
• C. \( \frac{x+1}{4}\sqrt{x^2+x+1} - \frac{3}{8}\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c \)
• D. \( \frac{(2x+1)}{4}\sqrt{x^2+x+1} - \frac{3}{8}\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c \)

Hint:

To integrate \( \int \sqrt{ax^2+bx+c} \, dx \): 1. Complete the square for the quadratic term \( ax^2+bx+c \) to bring it to the form \( A((x+h)^2 \pm k^2) \) or \( A(k^2 \pm (x+h)^2) \). 2. Use a substitution \( u = x+h \). 3. Apply standard integral formulas: \( \int \sqrt{u^2+a^2} \, du = \frac{u}{2}\sqrt{u^2+a^2} + \frac{a^2}{2}\log|u+\sqrt{u^2+a^2}| + C \) or \( \frac{u}{2}\sqrt{u^2+a^2} + \frac{a^2}{2}\sinh^{-1}\left(\frac{u}{a}\right) + C \). \( \int \sqrt{u^2-a^2} \, du = \frac{u}{2}\sqrt{u^2-a^2} - \frac{a^2}{2}\log|u+\sqrt{u^2-a^2}| + C \) or \( \frac{u}{2}\sqrt{u^2-a^2} - \frac{a^2}{2}\cosh^{-1}\left(\frac{u}{a}\right) + C \). \( \int \sqrt{a^2-u^2} \, du = \frac{u}{2}\sqrt{a^2-u^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{u}{a}\right) + C \).

Answer 1

The Correct Option is A

We need to evaluate \( I = \int \sqrt{x^2+x+1} \ dx \).
First, complete the square for the quadratic inside the square root: \( x^2+x+1 = \left(x^2+x+\frac{1}{4}\right) + 1 - \frac{1}{4} = \left(x+\frac{1}{2}\right)^2 + \frac{3}{4} \).
Let \( u = x+\frac{1}{2} \).
Then \( du = dx \).
The integral becomes \( I = \int \sqrt{u^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \ du \).
This is of the form \( \int \sqrt{u^2+a^2} \ du \), where \( a = \frac{\sqrt{3}}{2} \).
The standard integral formula is: \[ \int \sqrt{u^2+a^2} \ du = \frac{u}{2}\sqrt{u^2+a^2} + \frac{a^2}{2}\log|u+\sqrt{u^2+a^2}| + C \] Or, using inverse hyperbolic functions: \[ \int \sqrt{u^2+a^2} \ du = \frac{u}{2}\sqrt{u^2+a^2} + \frac{a^2}{2}\sinh^{-1}\left(\frac{u}{a}\right) + C \] Substitute \( u = x+\frac{1}{2} \) and \( a = \frac{\sqrt{3}}{2} \): \( a^2 = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \).
\[ I = \frac{x+1/2}{2}\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}} + \frac{3/4}{2}\sinh^{-1}\left(\frac{x+1/2}{\sqrt{3}/2}\right) + c \] \[ I = \frac{(2x+1)/2}{2}\sqrt{x^2+x+1} + \frac{3}{8}\sinh^{-1}\left(\frac{(2x+1)/2}{\sqrt{3}/2}\right) + c \] \[ I = \frac{2x+1}{4}\sqrt{x^2+x+1} + \frac{3}{8}\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right) + c \] This matches option (1).