Determine [tex]$\int y . dx$[/tex] if: [tex]$y=x \cdot e^{\frac{x}{10}}$[/tex]
Answer (2)
Recognize the need for integration by parts to solve ∫ x e 10 x d x .
Apply integration by parts with u = x and d v = e 10 x d x , leading to d u = d x and v = 10 e 10 x .
Use the formula ∫ u d v = uv − ∫ v d u to get ∫ x e 10 x d x = 10 x e 10 x − ∫ 10 e 10 x d x .
Evaluate the remaining integral and simplify to obtain the final answer: 10 e 10 x ( x − 10 ) + C .
Explanation
Problem Analysis We are asked to find the indefinite integral of the function y = x "." e 10 x with respect to x . This requires us to evaluate ∫ x e 10 x d x .
Applying Integration by Parts To solve this integral, we will use integration by parts. The formula for integration by parts is: ∫ u d v = uv − ∫ v d u We need to choose appropriate functions for u and d v . Let's choose u = x and d v = e 10 x d x .
Finding du and v Now, we find d u and v . d u = d x v = ∫ e 10 x d x = 10 e 10 x
Substituting into the Formula Substitute u , d v , d u , and v into the integration by parts formula: ∫ x e 10 x d x = x ( 10 e 10 x ) − ∫ 10 e 10 x d x
Evaluating the Remaining Integral Now, we need to evaluate the remaining integral: ∫ 10 e 10 x d x = 10 ∫ e 10 x d x = 10 ( 10 e 10 x ) = 100 e 10 x
Combining the Terms Substitute this back into the equation: ∫ x e 10 x d x = 10 x e 10 x − 100 e 10 x + C We can factor out 10 e 10 x from both terms: ∫ x e 10 x d x = 10 e 10 x ( x − 10 ) + C
Final Answer Therefore, the indefinite integral of x e 10 x is: ∫ x e 10 x d x = 10 e 10 x ( x − 10 ) + C
Examples
Imagine you're calculating the total drug concentration in a patient's bloodstream over time, where the drug's absorption rate is proportional to time and follows an exponential decay. The integral ∫ x e 10 x d x can model such scenarios, helping pharmacists determine optimal dosage and predict drug levels for effective treatment. This integration technique is crucial in pharmacokinetics for understanding drug behavior in the body.
To solve ∫ x e 10 x d x , we use integration by parts with u = x and d v = e 10 x d x . This results in the expression 10 e 10 x ( x − 10 ) + C after simplifying. Therefore, the final answer to the integral is 10 e 10 x ( x − 10 ) + C .
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