Evaluate the integral int frac{1}{(x+2)(x^2+1 | vedclass

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(B) To solve $\int \frac{1}{(x+2)(x^2+1)} \, dx$,we use partial fractions. Let $\frac{1}{(x+2)(x^2+1)} = \frac{A}{x+2} + \frac{Bx+C}{x^2+1}$. Multiply

Vedclass · Accepted Answer

$\frac{1}{5} \log \left| \frac{x+2}{\sqrt{x^2+1}} ight| + \frac{1}{5} 	an^{-1} x + c$

Vedclass · Answer

(B) To solve $\int \frac{1}{(x+2)(x^2+1)} \, dx$,we use partial fractions. Let $\frac{1}{(x+2)(x^2+1)} = \frac{A}{x+2} + \frac{Bx+C}{x^2+1}$. Multiplying by $(x+2)(x^2+1)$,we get $1 = A(x^2+1) + (Bx+C)(x+2)$. Setting $x = -2$,we get $1 = A(4+1) \implies A = \frac{1}{5}$. Comparing coefficients of $x^2$: $0 = A + B \implies B = -A = -\frac{1}{5}$. Comparing constants: $1 = A + 2C \implies 1 = \frac{1}{5} + 2C \implies 2C = \frac{4}{5} \implies C = \frac{2}{5}$. Thus,the integral becomes $\int \left( \frac{1/5}{x+2} + \frac{-1/5x + 2/5}{x^2+1} ight) \, dx$. $= \frac{1}{5} \int \frac{1}{x+2} \, dx - \frac{1}{10} \int \frac{2x}{x^2+1} \, dx + \frac{2}{5} \int \frac{1}{x^2+1} \, dx$. $= \frac{1}{5} \log |x+2| - \frac{1}{10} \log |x^2+1| + \frac{2}{5} 	an^{-1} x + c$. $= \frac{1}{5} \log \left| \frac{x+2}{\sqrt{x^2+1}} ight| + \frac{2}{5} 	an^{-1} x + c$.

Evaluate the integral $\int \frac{1}{(x+2)(x^2+1)} \, dx$.

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