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int(x^(2)+2x)/(x-3)dx

$\int \frac{x^{2}+2 x}{x-3} d x$

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Evaluate definite integrals using the chain rule

Full solution

Q.

\int \frac{x^{2}+2 x}{x-3} d x

Simplify using polynomial division: Step $1$ : Simplify the integrand using polynomial division. $\newline$ Divide $x^2 + 2x$ by $x - 3$ . $\newline$ $x^2 + 2x = (x - 3)(x + 5) + 15$ $\newline$ So, $(x^2 + 2x) / (x - 3) = x + 5 + \frac{15}{(x - 3)}$
Break into simpler parts: Step $2$ : Break the integral into simpler parts. \int\left(\frac{x^\(2\) + \(2\)x}{x - \(3\)}\right) dx = \int(x + \(5) dx + \int\left(\frac{ $15$ }{x - $3$ }\right) dx
Integrate each part: Step $3$ : Integrate each part. $\newline$ $\int(x + 5) \, dx = \frac{1}{2}x^2 + 5x + C$ $\newline$ $\int\frac{15}{(x - 3)} \, dx = 15 \ln|x - 3| + C$
Combine the results: Step $4$ : Combine the results from Step $3$ . $\newline$ $\int\left(\frac{x^2 + 2x}{x - 3}\right) dx = \frac{1}{2}x^2 + 5x + 15 \ln|x - 3| + C$

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