Evaluate the integral int(x-2)/(3x+6)dx. Choose 1 answer: (A) (1)/(2)x-(4ln |x+2|)/(3)+C (B) (1)/(2)x+(4ln |x+2|)/(3)+C (C) (1)/(3)x-(4ln |x+2|)/(3)+C (D) (1)/(3)x+(4ln |x+2|)/(3)+C

Rewrite Integral: Rewrite the integral by factoring out the constant from the denominator. int(x-2)/(3x+6)dx = int(x-2)/(3(x+2))dxSplit into Two: Split the integral into two separate integrals. int(x-2)/(3(x+2))dx = (1/3)int(x/(x+2))dx - (1/3)int(2/(x+2))dxSimplify First Integral: Simplify the first integral by dividing x by (x+2). (1/3)int(x/(x+2))dx = (1/3)int(1 - 2/(x+2))dxSeparate Integrals: Separate the integrals. (1/3)int(1 - 2/(x+2))dx = (1/3)int(1)dx - (2/3)int(1/(x+2))dxIntegrate Both Terms: Integrate both terms. (1/3)int(1)dx - (2/3)int(1/(x+2))dx = (1/3)x - (2/3)ln|x+2| + CMultiply by Constants: Multiply through by the constants. (1/3)x - (2/3)ln|x+2| + C = (1/3)x - (4/3)ln|x+2| + CMatch with Options: Match the result with the given options. The correct answer is (D) (1/3)x + (4/3)ln|x+2| + C

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Evaluate the integral
int(x-2)/(3x+6)dx.
Choose 1 answer:
(A)
(1)/(2)x-(4ln |x+2|)/(3)+C
(B)
(1)/(2)x+(4ln |x+2|)/(3)+C
(C)
(1)/(3)x-(4ln |x+2|)/(3)+C
(D)
(1)/(3)x+(4ln |x+2|)/(3)+C

Evaluate the integral $\int \frac{x-2}{3 x+6} d x$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{2} x-\frac{4 \ln |x+2|}{3}+C$ $\newline$ (B) $\frac{1}{2} x+\frac{4 \ln |x+2|}{3}+C$ $\newline$ (C) $\frac{1}{3} x-\frac{4 \ln |x+2|}{3}+C$ $\newline$ (D) $\frac{1}{3} x+\frac{4 \ln |x+2|}{3}+C$

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate the integral

\int \frac{x-2}{3 x+6} d x

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{2} x-\frac{4 \ln |x+2|}{3}+C

\newline

(B)

\frac{1}{2} x+\frac{4 \ln |x+2|}{3}+C

\newline

(C)

\frac{1}{3} x-\frac{4 \ln |x+2|}{3}+C

\newline

(D)

\frac{1}{3} x+\frac{4 \ln |x+2|}{3}+C

Rewrite Integral: Rewrite the integral by factoring out the constant from the denominator. \int\frac{x$-2$}{$3$x+$6$}\,dx = \int\frac{x$-2$}{$3$(x+$2$)}\,dx
Split into Two: Split the integral into two separate integrals. \(\int\frac{x-2}{3(x+2)}dx = \frac{1}{3}\int\frac{x}{x+2}dx - \frac{1}{3}\int\frac{2}{x+2}dx
Simplify First Integral: Simplify the first integral by dividing $x$ by $(x+2)$ . $\frac{1}{3}\int\frac{x}{(x+2)}dx = \frac{1}{3}\int(1 - \frac{2}{(x+2)})dx$
Separate Integrals: Separate the integrals. $\newline$ (\frac{$1$}{$3$})\int(\(1 - \frac{ $2$ }{x+ $2$ })dx = (\frac{ $1$ }{ $3$ })\int( $1$ )dx - (\frac{ $2$ }{ $3$ })\int(\frac{ $1$ }{x+ $2$ })dx
Integrate Both Terms: Integrate both terms. $\newline$ (\frac{$1$}{$3$})\int(\(1)dx - (\frac{ $2$ }{ $3$ })\int(\frac{ $1$ }{(x+ $2$ )})dx = (\frac{ $1$ }{ $3$ })x - (\frac{ $2$ }{ $3$ })\ln|x+ $2$ | + C
Multiply by Constants: Multiply through by the constants. $\newline$ (\frac{$1$}{$3$})x - (\frac{$2$}{$3$})\ln|x+\(2| + C = (\frac{ $1$ }{ $3$ })x - (\frac{ $4$ }{ $3$ })\ln|x+ $2$ | + C
Match with Options: Match the result with the given options. $\newline$ The correct answer is (D) $(\frac{1}{3})x + (\frac{4}{3})\ln|x+2| + C$