$\int \frac{1}{(x+1)^2+4} d x=$
A. $\frac{1}{2} \arctan \left(\frac{x+1}{2}\right)+C$
B. $\frac{1}{4} \arctan (x+1)+C$
C. $\frac{1}{2} \arctan (1-x)+C$
D. $\frac{1}{4} \arctan \left(\frac{x+1}{4}\right)+C$
E. $\arctan \left(\frac{x+1}{2}\right)+C$
Answer (2)
Perform a u-substitution: Let u = x + 1 , so d u = d x , and the integral becomes ∫ u 2 + 4 1 d u .
Apply the standard integral formula: ∫ u 2 + a 2 1 d u = a 1 arctan ( a u ) + C , where a = 2 , resulting in 2 1 arctan ( 2 u ) + C .
Substitute back u = x + 1 : This gives 2 1 arctan ( 2 x + 1 ) + C .
The final answer is: 2 1 arctan ( 2 x + 1 ) + C
Explanation
Problem Analysis We are asked to evaluate the indefinite integral ∫ ( x + 1 ) 2 + 4 1 d x and choose the correct answer from the given options.
U-Substitution Let's use a u -substitution to simplify the integral. Let u = x + 1 , then d u = d x . The integral becomes ∫ u 2 + 4 1 d u .
Applying the Formula Now we can use the 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 + 1 to get the final answer: 2 1 arctan ( 2 x + 1 ) + C .
Comparing with Options Comparing our result with the given options, we see that option a matches our answer.
Examples
Imagine you are designing a suspension bridge and need to calculate the sag curve. The integral you just solved can model the shape of the cable under certain load conditions. Understanding how to solve such integrals allows engineers to accurately predict the behavior of structures and ensure their stability.
The integral ∫ ( x + 1 ) 2 + 4 1 d x can be solved using u-substitution, leading to 2 1 arctan ( 2 x + 1 ) + C . The final answer corresponds to option A. Thus, the solution is 2 1 arctan ( 2 x + 1 ) + C .
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