Find the anti-derivative and simplify.
[tex]\begin{array}{l}
\int \frac{x^2}{x^6+16} d x \\
\frac{\tan ^{-1}\left(\frac{x^3}{4}\right)}{[?]}+C
\end{array}[/tex]
Answer (2)
To find the anti-derivative of ∫ x 6 + 16 x 2 d x , we use u-substitution with u = x 3 and arrive at 12 t a n − 1 ( 4 x 3 ) + C . The missing term is 12.
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Rewrite the integral as ∫ ( x 3 ) 2 + 4 2 x 2 d x .
Use u-substitution: let u = x 3 , so d u = 3 x 2 d x , and x 2 d x = 3 1 d u .
Apply the formula ∫ x 2 + a 2 1 d x = a 1 tan − 1 ( a x ) + C and substitute back u = x 3 .
The antiderivative is 12 t a n − 1 ( 4 x 3 ) + C , so the missing term is 12 .
Explanation
Problem Analysis We are asked to find the antiderivative of the function x 6 + 16 x 2 and express the result in the form [ ?] t a n − 1 ( 4 x 3 ) + C , where we need to determine the missing term.
U-Substitution To find the antiderivative, we can rewrite the integral as ∫ ( x 3 ) 2 + 4 2 x 2 d x . This form suggests a u-substitution. Let u = x 3 , then d u = 3 x 2 d x , which means x 2 d x = 3 1 d u .
Substituting u Now, substitute u into the integral: ∫ x 6 + 16 x 2 d x = ∫ u 2 + 16 1 ⋅ 3 1 d u = 3 1 ∫ u 2 + 4 2 1 d u
Applying the Formula We know that ∫ x 2 + a 2 1 d x = a 1 tan − 1 ( a x ) + C . Applying this formula, we get: 3 1 ∫ u 2 + 4 2 1 d u = 3 1 ⋅ 4 1 tan − 1 ( 4 u ) + C = 12 1 tan − 1 ( 4 u ) + C
Substituting Back Now, substitute back u = x 3 to get the antiderivative in terms of x : 12 1 tan − 1 ( 4 x 3 ) + C
Finding the Missing Term Comparing this with the given form [ ?] t a n − 1 ( 4 x 3 ) + C , we see that the missing term is 12.
Final Answer Therefore, the antiderivative is 12 t a n − 1 ( 4 x 3 ) + C .
Examples
Imagine you're designing a cooling system for a computer processor. The rate at which heat dissipates might be modeled by a function similar to the one in this problem. Finding the antiderivative helps you determine the total heat dissipated over a period, which is crucial for designing an effective cooling solution and preventing overheating.
