Maths · Evaluation of definite integrals
= \)
\( \boldsymbol{n} \stackrel{L t}{\rightarrow} \infty\left[\frac{\boldsymbol{n}+\mathbf{1}}{\boldsymbol{n}^{2}+\mathbf{1}^{2}}+\frac{\boldsymbol{n}+\boldsymbol{2}}{\boldsymbol{n}^{2}+\mathbf{2}^{2}}+\ldots+\right. \) \( \left.\frac{\boldsymbol{n}+\boldsymbol{n}}{\boldsymbol{n}^{2}+\boldsymbol{n}^{2}}\right]= \)
- A. \( \frac{\pi}{4}+\frac{1}{2} \log 2 \)
- B. \( \frac{\pi}{4}-\frac{1}{2} \log 2 \)
- C. \( \frac{\pi}{2}+\frac{1}{2} \log 2 \)
- D. \( \frac{\pi}{4}+\frac{1}{4} \log 2 \)
Step-by-step solution
Rewrite the sum as (1/n) Σ (1 + k/n)/(1 + (k/n)^2). As n→∞, it becomes ∫_0^1 (1+x)/(1+x^2) dx = ∫_0^1 dx/(1+x^2) + ∫_0^1 x/(1+x^2) dx = arctan x|_0^1 + (1/2) ln(1+x^2)|_0^1 = π/4 + (1/2) ln 2.