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Evaluate
int_(11)^(e^(3)+10)(2x-19)/(x-10)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Evaluate $\int_{11}^{e^{3}+10} \frac{2 x-19}{x-10} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate

\int_{11}^{e^{3}+10} \frac{2 x-19}{x-10} d x

. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Simplify: Simplify the integrand if possible. $\newline$ The integrand $(2x-19)/(x-10)$ can be simplified by long division since the degree of the numerator is equal to the degree of the denominator. We divide $2x$ by $x$ to get $2$ and multiply $(x-10)$ by $2$ to get $2x-20$ . We then subtract this from the numerator to get a remainder of $1$ . So, the integrand simplifies to $2 + 1/(x-10)$ .
Break down: Break the integral into two simpler integrals. $\newline$ We can write the integral as the sum of two simpler integrals: $\newline$ $\int\frac{2x-19}{x-10} dx = \int 2 dx + \int\frac{1}{x-10} dx$
Integrate separately: Integrate each term separately. $\newline$ The integral of $2$ with respect to $x$ is $2x$ , and the integral of $1/(x-10)$ with respect to $x$ is $\ln|x-10|$ . So we have: $\newline$ $\int 2 \, dx = 2x$ $\newline$ $\int \frac{1}{(x-10)} \, dx = \ln|x-10|$
Combine and evaluate: Combine the antiderivatives and evaluate the definite integral. $\newline$ The combined antiderivative is $2x + \ln|x-10|$ . We need to evaluate this from $11$ to $e^{3}+10$ : $\newline$ $(2x + \ln|x-10|)$ evaluated from $11$ to $e^{3}+10$ is: $\newline$ $(2(e^{3}+10) + \ln|e^{3}+10-10|) - (2(11) + \ln|11-10|)$
Simplify expression: Simplify the expression. $\newline$ Now we plug in the limits of integration: $\newline$ $= (2(e^{3}+10) + \ln|e^{3}|) - (22 + \ln|1|)$ $\newline$ $= (2e^{3} + 20 + \ln(e^{3})) - (22 + \ln(1))$ $\newline$ Since $\ln(e^{3}) = 3$ and $\ln(1) = 0$ , we can further simplify: $\newline$ $= (2e^{3} + 20 + 3) - (22 + 0)$ $\newline$ $= 2e^{3} + 23 - 22$ $\newline$ $= 2e^{3} + 1$

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