$\int \frac{4}{\sqrt{-6 x^2-24 x+6}} d x$
A. $\frac{2}{\sqrt{6}} \arcsin \left(\frac{x-2}{\sqrt{6}}\right)+C$
B. $\frac{4}{\sqrt{6}} \arcsin \left(\frac{x+2}{\sqrt{5}}\right)+C$
C. $\frac{2}{\sqrt{6}} \arcsin (z+\sqrt{5})+C$
D. $\frac{4}{\sqrt{ } 5} \arcsin \left(\frac{z+2}{\sqrt{ } 6}\right)+C$
E. $\sqrt{6} \arcsin (x+2)+C$
Answer (2)
The integral ∫ − 6 x 2 − 24 x + 6 4 d x evaluates to 6 4 arcsin ( 5 x + 2 ) + C . The correct answer is option B. This process involved completing the square, substitution, and applying the arcsine formula.
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Complete the square: − 6 x 2 − 24 x + 6 = 6 ( 5 − ( x + 2 ) 2 ) .
Rewrite the integral: 6 4 ∫ 5 − ( x + 2 ) 2 1 d x .
Use u-substitution: u = x + 2 , resulting in 6 4 ∫ 5 − u 2 1 d u .
Apply arcsin formula and substitute back: 6 4 arcsin ( 5 x + 2 ) + C . The final answer is 6 4 arcsin ( 5 x + 2 ) + C .
Explanation
Problem Analysis We are given the integral ∫ − 6 x 2 − 24 x + 6 4 d x and asked to find its solution from the given options.
Completing the Square First, we complete the square for the quadratic expression inside the square root:
− 6 x 2 − 24 x + 6 = − 6 ( x 2 + 4 x ) + 6 = − 6 ( x 2 + 4 x + 4 − 4 ) + 6 = − 6 (( x + 2 ) 2 − 4 ) + 6 = − 6 ( x + 2 ) 2 + 24 + 6 = 30 − 6 ( x + 2 ) 2 = 6 ( 5 − ( x + 2 ) 2 ) .
Rewriting the Integral Now we rewrite the integral using the completed square form:
∫ 6 ( 5 − ( x + 2 ) 2 ) 4 d x = 6 4 ∫ 5 − ( x + 2 ) 2 1 d x .
U-Substitution Next, we perform a u-substitution: Let u = x + 2 , then d u = d x . The integral becomes
6 4 ∫ 5 − u 2 1 d u .
Applying Arcsin Integral Formula Now we apply the arcsin integral formula: ∫ a 2 − u 2 1 d u = arcsin ( a u ) + C . In this case, a 2 = 5 , so a = 5 . The integral becomes
6 4 arcsin ( 5 u ) + C .
Substituting Back Finally, we substitute back u = x + 2 :
6 4 arcsin ( 5 x + 2 ) + C .
Final Answer Comparing the result with the given options, we see that option b matches our solution.
Therefore, the answer is 6 4 arcsin ( 5 x + 2 ) + C .
Examples
Imagine you're designing a suspension bridge and need to calculate the sag of the cable. The integral you solved is similar to those used to determine the curve of the cable under load. By understanding how to evaluate such integrals, you can accurately predict the shape and stability of the bridge, ensuring its safety and efficiency. This showcases how calculus is essential in structural engineering for designing safe and reliable infrastructure.
