Evaluate int(x^(2)+2x+3)/(x+1)dx

Simplify the integrand: Simplify the integrand by performing polynomial long division or synthetic division. We have the integral of a rational function, where the numerator is a polynomial of degree 2 and the denominator is a polynomial of degree 1. We can simplify the integrand by dividing x^2 + 2x + 3 by x + 1. Performing the division, we get: (x^2 + 2x + 3) / (x + 1) = x + 1 + 2 / (x + 1)Rewrite with simplified integrand: Rewrite the integral with the simplified integrand. Now we can write the integral as the sum of two simpler integrals: ∫(x^2 + 2x + 3) / (x + 1) dx = ∫(x + 1) dx + ∫2 / (x + 1) dxIntegrate first part: Integrate the first part of the sum. The integral of x + 1 with respect to x is straightforward: ∫(x + 1) dx = (1/2)x^2 + x + C_1 where C_1 is the constant of integration.Integrate second part: Integrate the second part of the sum. The integral of 2 / (x + 1) with respect to x is a standard integral that results in a natural logarithm: ∫2 / (x + 1) dx = 2ln|x + 1| + C_2 where C_2 is another constant of integration.Combine results: Combine the results from Step 3 and Step 4. Adding the two integrals together, we get the final result: ∫(x^2 + 2x + 3) / (x + 1) dx = (1/2)x^2 + x + 2ln|x + 1| + C where C is the combined constant of integration (C = C_1 + C_2).

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Evaluate $\int\frac{x^{2}+2x+3}{x+1}\,dx$

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Evaluate definite integrals using the chain rule

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Q. Evaluate

\int\frac{x^{2}+2x+3}{x+1}\,dx

Simplify the integrand: Simplify the integrand by performing polynomial long division or synthetic division. $\newline$ We have the integral of a rational function, where the numerator is a polynomial of degree $2$ and the denominator is a polynomial of degree $1$ . We can simplify the integrand by dividing $x^2 + 2x + 3$ by $x + 1$ . $\newline$ Performing the division, we get: $\newline$ $(x^2 + 2x + 3) / (x + 1) = x + 1 + \frac{2}{x + 1}$
Rewrite with simplified integrand: Rewrite the integral with the simplified integrand. $\newline$ Now we can write the integral as the sum of two simpler integrals: $\newline$ $\int\frac{x^2 + 2x + 3}{x + 1} dx = \int(x + 1) dx + \int\frac{2}{x + 1} dx$
Integrate first part: Integrate the first part of the sum. $\newline$ The integral of $x + 1$ with respect to $x$ is straightforward: $\newline$ $\int(x + 1) dx = \frac{1}{2}x^2 + x + C_1$ $\newline$ where $C_1$ is the constant of integration.
Integrate second part: Integrate the second part of the sum. $\newline$ The integral of $2 / (x + 1)$ with respect to $x$ is a standard integral that results in a natural logarithm: $\newline$ $\int \frac{2}{(x + 1)} dx = 2\ln|x + 1| + C_2$ $\newline$ where $C_2$ is another constant of integration.
Combine results: Combine the results from Step $3$ and Step $4$ . $\newline$ Adding the two integrals together, we get the final result: $\newline$ $\int(\frac{x^2 + 2x + 3}{x + 1}) \, dx = (\frac{1}{2})x^2 + x + 2\ln|x + 1| + C$ $\newline$ where $C$ is the combined constant of integration ( $C = C_1 + C_2$ ).