$\int \frac{2 x+1}{4+x^2} d x$
Answer (2)
Split the integral into two parts: ∫ 4 + x 2 2 x d x + ∫ 4 + x 2 1 d x .
Use u-substitution for the first integral, where u = 4 + x 2 , resulting in ln ( 4 + x 2 ) .
Recognize the second integral as a standard arctangent form, giving 2 1 arctan ( 2 x ) .
Combine the results to get the final answer: ln ( 4 + x 2 ) + 2 1 arctan ( 2 x ) + C
Explanation
Splitting the Integral We are asked to find the indefinite integral of the function 4 + x 2 2 x + 1 . To solve this, we can split the integral into two parts: one involving 4 + x 2 2 x and the other involving 4 + x 2 1 .
Rewriting the Integral We can rewrite the integral as: ∫ 4 + x 2 2 x + 1 d x = ∫ 4 + x 2 2 x d x + ∫ 4 + x 2 1 d x
Solving the First Integral For the first integral, ∫ 4 + x 2 2 x d x , we can use a simple u-substitution. Let u = 4 + x 2 , then d u = 2 x d x . The integral becomes: ∫ u 1 d u = ln ∣ u ∣ + C 1 = ln ∣4 + x 2 ∣ + C 1 Since 4 + x 2 is always positive, we can drop the absolute value: ln ( 4 + x 2 ) + C 1
Solving the Second Integral For the second integral, ∫ 4 + x 2 1 d x , we recognize this as a standard form integral of the type ∫ a 2 + x 2 1 d x = a 1 arctan ( a x ) + C , where a = 2 . Thus, we have: ∫ 4 + x 2 1 d x = 2 1 arctan ( 2 x ) + C 2
Combining the Results Now, we combine the results of the two integrals: ∫ 4 + x 2 2 x + 1 d x = ln ( 4 + x 2 ) + 2 1 arctan ( 2 x ) + C where C = C 1 + C 2 is the constant of integration.
Final Answer Therefore, the indefinite integral of 4 + x 2 2 x + 1 is ln ( 4 + x 2 ) + 2 1 arctan ( 2 x ) + C .
Examples
Imagine you are calculating the total amount of drug concentration in the bloodstream over time, where the rate of absorption and elimination is described by a function similar to the one in this problem. The integral helps determine the cumulative effect of the drug, crucial for determining dosage and treatment effectiveness. Understanding how to solve such integrals is essential in pharmacology and biomedical engineering.
To solve the integral ∫ 4 + x 2 2 x + 1 d x , we split it into two parts and solve each separately. The final answer combines the results as ln ( 4 + x 2 ) + 2 1 arctan ( 2 x ) + C . This method utilizes u-substitution and the standard form of the arctangent integral.
;
