Find int sqrt{x^{2}+2 x+5} , dx. | vedclass

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Question

(A) We need to evaluate the integral $I = \int \sqrt{x^{2}+2 x+5} \, dx$. First,complete the square for the quadratic expression: $x^{2}+2 x+5 = (x^{2

Vedclass · Accepted Answer

$\frac{1}{2}(x+1) \sqrt{x^{2}+2 x+5}+2 \log |x+1+\sqrt{x^{2}+2 x+5}|+ C$

Vedclass · Answer

(A) We need to evaluate the integral $I = \int \sqrt{x^{2}+2 x+5} \, dx$. First,complete the square for the quadratic expression: $x^{2}+2 x+5 = (x^{2}+2 x+1) + 4 = (x+1)^{2} + 2^{2}$. Now,the integral becomes $I = \int \sqrt{(x+1)^{2} + 2^{2}} \, dx$. Using the standard formula $\int \sqrt{t^{2}+a^{2}} \, dt = \frac{1}{2} t \sqrt{t^{2}+a^{2}} + \frac{a^{2}}{2} \log |t + \sqrt{t^{2}+a^{2}}| + C$,where $t = x+1$ and $a = 2$: $I = \frac{1}{2}(x+1) \sqrt{(x+1)^{2} + 2^{2}} + \frac{2^{2}}{2} \log |(x+1) + \sqrt{(x+1)^{2} + 2^{2}}| + C$. Simplifying the expression: $I = \frac{1}{2}(x+1) \sqrt{x^{2}+2 x+5} + 2 \log |x+1 + \sqrt{x^{2}+2 x+5}| + C$.

Find $\int \sqrt{x^{2}+2 x+5} \, dx$.

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