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

Split Integral: Let's split the integral into two parts by dividing each term in the numerator by the denominator. \[ \int \frac{x-1}{2x+4}dx = \int \frac{1}{2} - \frac{1}{2x+4}dx \]Integrate Parts: Now we integrate each part separately. \[ \int \frac{1}{2}dx = \frac{1}{2}x \] and \[ \int \frac{1}{2x+4}dx = \frac{1}{2} \int \frac{1}{x+2}dx \]Use Natural Logarithm: To integrate $\frac{1}{x+2}$, we use the natural logarithm. \[ \frac{1}{2} \int \frac{1}{x+2}dx = \frac{1}{2} \ln|x+2| + C \]Combine Parts: Combine the two parts to get the final answer. \[ \frac{1}{2}x - \frac{1}{2} \ln|x+2| + C \]Match Final Answer: Match the final answer with the given options. The correct answer is (A) $\frac{1}{2}x - \frac{1}{2}\ln|x+2| + C$.

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

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

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Math Problems

Calculus

Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate the integral

\int \frac{x-1}{2 x+4} d x

\newline

Choose

1

answer:

\newline

(A)

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

\newline

(B)

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

\newline

(C)

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

\newline

(D)

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

Split Integral: Let's split the integral into two parts by dividing each term in the numerator by the denominator. $\newline$ $\int \frac{x-1}{2x+4}dx = \int \frac{1}{2} - \frac{1}{2x+4}dx$
Integrate Parts: Now we integrate each part separately. $\newline$ $\int \frac{1}{2}dx = \frac{1}{2}x$ $\newline$ and $\newline$ $\int \frac{1}{2x+4}dx = \frac{1}{2} \int \frac{1}{x+2}dx$
Use Natural Logarithm: To integrate $\frac{1}{x+2}$ , we use the natural logarithm. $\newline$ $\frac{1}{2} \int \frac{1}{x+2}dx = \frac{1}{2} \ln|x+2| + C$
Combine Parts: Combine the two parts to get the final answer. $\newline$ $\frac{1}{2}x - \frac{1}{2} \ln|x+2| + C$
Match Final Answer: Match the final answer with the given options. $\newline$ The correct answer is (A) $\frac{1}{2}x - \frac{1}{2}\ln|x+2| + C$ .