$\int \frac{1}{x^2+6 x+11} d x=$
A. $\frac{1}{2} \arctan \left(\frac{x-3}{\sqrt{2}}\right)+C$
B. $\frac{\sqrt{2}}{2} \arctan \left(\frac{x+3}{\sqrt{2}}\right)+C$
C. $\frac{1}{2} \arctan \left(\frac{x+3}{2}\right)+C$
D. $\frac{\sqrt{2}}{2} \arctan (x+3)+C$
E. $\frac{1}{2} \arctan \left(\frac{x+3}{\sqrt{2}}\right)+C$
Answer (2)
The integral ∫ x 2 + 6 x + 11 1 d x is solved by completing the square to transform the denominator, applying a substitution, and using the arctangent integration formula. The final answer is 2 2 arctan ( 2 x + 3 ) + C , which matches option B from the provided choices.
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Complete the square in the denominator: x 2 + 6 x + 11 = ( x + 3 ) 2 + 2 .
Use the substitution u = x + 3 , so d u = d x . The integral becomes ∫ u 2 + 2 1 d u .
Apply the formula ∫ u 2 + a 2 1 d u = a 1 arctan ( a u ) + C , where a = 2 , to get 2 1 arctan ( 2 u ) + C .
Substitute back u = x + 3 and rationalize the denominator to get the final answer: 2 2 arctan ( 2 x + 3 ) + C .
Explanation
Problem Analysis We are asked to find the indefinite integral of the function x 2 + 6 x + 11 1 . We will solve this by completing the square in the denominator and then using a standard integral formula.
Completing the Square First, complete the square in the denominator:
x 2 + 6 x + 11 = ( x 2 + 6 x + 9 ) + 11 − 9 = ( x + 3 ) 2 + 2
Rewriting the Integral Now, rewrite the integral using the completed square form:
∫ x 2 + 6 x + 11 1 d x = ∫ ( x + 3 ) 2 + 2 1 d x
U-Substitution Use the substitution u = x + 3 , so d u = d x . The integral becomes:
∫ u 2 + 2 1 d u
Applying the Formula Now, we use the standard integral formula ∫ u 2 + a 2 1 d u = a 1 arctan ( a u ) + C , where a = 2 . Applying this formula, we get:
2 1 arctan ( 2 u ) + C
Substituting Back Substitute back u = x + 3 to get:
2 1 arctan ( 2 x + 3 ) + C
Rationalizing the Denominator Rationalize the denominator:
2 1 arctan ( 2 x + 3 ) + C = 2 2 arctan ( 2 x + 3 ) + C
Final Answer Comparing the result with the given options, we find that the correct answer is:
2 2 arctan ( 2 x + 3 ) + C
Selecting the Correct Option Therefore, the correct option is b.
Examples
The integral of the form ∫ x 2 + 6 x + 11 1 d x can be used to model various phenomena in physics and engineering. For example, it can represent the response of a damped harmonic oscillator to an external force. In circuit analysis, such integrals appear when calculating the current or voltage in circuits with resistors, inductors, and capacitors. Understanding how to solve these integrals allows engineers to predict and control the behavior of these systems.
