Consider the indefinite integral $\int \frac{-8 e^{-8 x}}{\left(e^{-8 x}+3\right)^5} d x$:
This can be transformed into a basic integral by letting
$u=$ $\square$ and
$d u=\square d x$
Performing the substitution yields the integral
Answer (2)
Let u = e − 8 x + 3 .
Find d u = − 8 e − 8 x d x .
Substitute u and d u into the integral.
The integral becomes ∫ u − 5 d u , so the answer is ∫ u − 5 d u .
Explanation
Problem Analysis We are given the indefinite integral: ∫ ( e − 8 x + 3 ) 5 − 8 e − 8 x d x Our goal is to transform this integral into a basic integral using u-substitution.
Choosing the Substitution Let's choose a suitable substitution. A good choice is to let u = e − 8 x + 3 This is because the derivative of e − 8 x is − 8 e − 8 x , which appears in the numerator of the integrand.
Finding du Now, we compute the derivative of u with respect to x :
d x d u = − 8 e − 8 x Thus, d u = − 8 e − 8 x d x This matches the numerator of the original integral.
Performing the Substitution Now we substitute u and d u into the original integral: ∫ ( e − 8 x + 3 ) 5 − 8 e − 8 x d x = ∫ u 5 1 d u We can rewrite this as: ∫ u − 5 d u
Final Answer Therefore, the indefinite integral can be transformed into a basic integral by letting u = e − 8 x + 3 d u = − 8 e − 8 x d x and the resulting integral is ∫ u − 5 d u
Examples
U-substitution is a powerful technique used in physics to solve problems involving related rates. For example, when analyzing the motion of a damped harmonic oscillator, the equation often involves exponential decay terms. By using u-substitution, physicists can simplify the integral and find the displacement of the oscillator as a function of time, allowing them to predict the system's behavior.
The indefinite integral can be transformed by using the substitution u = e − 8 x + 3 , which simplifies the integral to ∫ u − 5 d u . This allows us to easily integrate the function. After integrating, we substitute back to find the original variables.
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