Evaluate the integral $\int_4^5 \frac{2}{x^2-8 x+17} d x$.
A. $\frac{\pi}{2}$
B. 0
C. $\frac{\pi}{4}$
D. 1
E. $\frac{\pi}{3}$
Answer (2)
The evaluation of the integral ∫ 4 5 x 2 − 8 x + 17 2 d x simplifies to 2 π after completing the square and using substitution. Thus, the answer is 2 π , which matches option A.
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Complete the square in the denominator: x 2 − 8 x + 17 = ( x − 4 ) 2 + 1 .
Substitute u = x − 4 , changing the limits of integration from x = 4 to u = 0 and x = 5 to u = 1 .
Evaluate the integral: 2 ∫ 0 1 u 2 + 1 1 d u = 2 [ arctan ( u ) ] 0 1 = 2 ( arctan ( 1 ) − arctan ( 0 )) = 2 ( 4 π − 0 ) .
Simplify to find the final answer: 2 π .
Explanation
Problem Analysis We are asked to evaluate the definite integral ∫ 4 5 x 2 − 8 x + 17 2 d x . To solve this, we will complete the square in the denominator and then use a trigonometric substitution.
Completing the Square First, complete the square in the denominator:
x 2 − 8 x + 17 = ( x 2 − 8 x + 16 ) + 1 = ( x − 4 ) 2 + 1
So the integral becomes:
∫ 4 5 ( x − 4 ) 2 + 1 2 d x
U-Substitution Now, we make a substitution: let u = x − 4 , so d u = d x . When x = 4 , u = 0 , and when x = 5 , u = 1 . The integral in terms of u is:
∫ 0 1 u 2 + 1 2 d u
Evaluating the Integral We know that ∫ u 2 + 1 1 d u = arctan ( u ) + C , so we have:
2 ∫ 0 1 u 2 + 1 1 d u = 2 [ arctan ( u ) ] 0 1 = 2 ( arctan ( 1 ) − arctan ( 0 ))
Since arctan ( 1 ) = 4 π and arctan ( 0 ) = 0 , the result is:
2 ( 4 π − 0 ) = 2 π
Final Answer Therefore, the value of the integral is 2 π .
Examples
Imagine you are calculating the average value of a function over an interval. This involves evaluating a definite integral, much like the one we solved. For instance, if you're analyzing the behavior of a circuit's voltage over time, you might need to integrate a function representing the voltage to find its average value. This skill is crucial in electrical engineering and signal processing.
