$\int \frac{2}{(x-2)^2+25} d x=$
A. $\frac{5}{2} \arctan \left(\frac{x-2}{5}\right)+C$
B. $\frac{2}{5} \arctan \left(\frac{x-2}{25}\right)+C$
C. $\frac{1}{25} \arctan (x-2)+C$
D. $\frac{2}{5} \arctan \left(\frac{x-2}{5}\right)+C$
E. $\frac{1}{5} \arctan (x-2)+C$
Answer (2)
The solution to the integral ∫ ( x − 2 ) 2 + 25 2 d x is 5 2 arctan ( 5 x − 2 ) + C . Thus, the correct multiple choice answer is D. This result is obtained by using a substitution followed by applying the arctangent integral formula.
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Perform a u-substitution: u = x − 2 , d u = d x .
Rewrite the integral: 2 ∫ u 2 + 5 2 1 d u .
Apply the formula: ∫ u 2 + a 2 1 d u = a 1 arctan ( a u ) + C .
Substitute back to obtain the final answer: 5 2 arctan ( 5 x − 2 ) + C .
Explanation
Problem Analysis We are given the integral ∫ ( x − 2 ) 2 + 25 2 d x and asked to find its solution from the given options. This integral can be solved using a simple substitution and the standard integral formula for arctan .
Substitution Let's use the substitution u = x − 2 . Then, d u = d x . The integral becomes: ∫ u 2 + 25 2 d u = 2 ∫ u 2 + 5 2 1 d u
Applying the Formula We know that ∫ u 2 + a 2 1 d u = a 1 arctan ( a u ) + C . In our case, a = 5 . So, 2 ∫ u 2 + 5 2 1 d u = 2 ⋅ 5 1 arctan ( 5 u ) + C = 5 2 arctan ( 5 u ) + C
Substituting Back Now, substitute back u = x − 2 to get: 5 2 arctan ( 5 x − 2 ) + C
Final Answer Comparing our result with the given options, we find that option d matches our solution. Therefore, the correct answer is: 5 2 arctan ( 5 x − 2 ) + C
Examples
Imagine you are designing a cooling system for an electronic device. The heat dissipation might be modeled by a function that involves a term similar to the integrand in this problem. Evaluating such integrals helps determine the temperature distribution and optimize the cooling efficiency, ensuring the device operates within safe temperature limits. This is crucial for preventing overheating and ensuring the longevity of the electronic components.
