int(3x^(3)-5x^(2)+10 x-3)/(3x+1)dx= Choose 1 answer: (A) (x^(3))/(3)+x^(2)+2x-7ln |3x+1|+C (B) (x^(3))/(3)-x^(2)+2x-(7ln |3x+1|)/(3)+C (C) (x^(3))/(3)+(x^(2))/(2)+4x-(7ln |3x+1|)/(3)+C (D) (x^(3))/(3)-x^(2)+4x-(7ln |3x+1|)/(3)+C

Polynomial Long Division: First, let's try polynomial long division to simplify the integrand.Integration Steps: Divide 3x^3 by 3x to get x^2. Multiply (3x+1) by x^2 to get 3x^3+x^2. Subtract this from the original polynomial to get -6x^2+10x-3.Step 1: Now, divide -6x^2 by 3x to get -2x. Multiply (3x+1) by -2x to get -6x^2-2x. Subtract this from the previous remainder to get 12x-3.Step 2: Divide 12x by 3x to get 4. Multiply (3x+1) by 4 to get 12x+4. Subtract this from the previous remainder to get -7.Step 3: So, the polynomial long division gives us x^2 - 2x + 4 - 7/(3x+1).Step 4: Now, integrate each term separately: ∫x^2 dx, ∫-2x dx, ∫4 dx, and ∫-7/(3x+1) dx.Step 5: The integral of x^2 is (1/3)x^3.Step 6: The integral of -2x is -x^2.Step 7: The integral of 4 is 4x.Step 8: The integral of -7/(3x+1) is -7ln|3x+1|, because the derivative of 3x+1 is 3, and we have to divide by that coefficient.Step 9: So, the integral of the original function is (1/3)x^3 - x^2 + 4x - (7/3)ln|3x+1| + C.Final Integration: Comparing with the answer choices, it looks like the correct answer is (D).

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int(3x^(3)-5x^(2)+10 x-3)/(3x+1)dx=
Choose 1 answer:
(A)
(x^(3))/(3)+x^(2)+2x-7ln |3x+1|+C
(B)
(x^(3))/(3)-x^(2)+2x-(7ln |3x+1|)/(3)+C
(C)
(x^(3))/(3)+(x^(2))/(2)+4x-(7ln |3x+1|)/(3)+C
(D)
(x^(3))/(3)-x^(2)+4x-(7ln |3x+1|)/(3)+C

$\int \frac{3 x^{3}-5 x^{2}+10 x-3}{3 x+1} d x=$ $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{x^{3}}{3}+x^{2}+2 x-7 \ln |3 x+1|+C$ $\newline$ (B) $\frac{x^{3}}{3}-x^{2}+2 x-\frac{7 \ln |3 x+1|}{3}+C$ $\newline$ (C) $\frac{x^{3}}{3}+\frac{x^{2}}{2}+4 x-\frac{7 \ln |3 x+1|}{3}+C$ $\newline$ (D) $\frac{x^{3}}{3}-x^{2}+4 x-\frac{7 \ln |3 x+1|}{3}+C$

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\int \frac{3 x^{3}-5 x^{2}+10 x-3}{3 x+1} d x=

\newline

Choose

1

answer:

\newline

(A)

\frac{x^{3}}{3}+x^{2}+2 x-7 \ln |3 x+1|+C

\newline

(B)

\frac{x^{3}}{3}-x^{2}+2 x-\frac{7 \ln |3 x+1|}{3}+C

\newline

(C)

\frac{x^{3}}{3}+\frac{x^{2}}{2}+4 x-\frac{7 \ln |3 x+1|}{3}+C

\newline

(D)

\frac{x^{3}}{3}-x^{2}+4 x-\frac{7 \ln |3 x+1|}{3}+C

Polynomial Long Division: First, let's try polynomial long division to simplify the integrand.
Integration Steps: Divide $3x^3$ by $3x$ to get $x^2$ . Multiply $(3x+1)$ by $x^2$ to get $3x^3+x^2$ . Subtract this from the original polynomial to get $-6x^2+10x-3$ .
Step $1$ : Now, divide $-6x^2$ by $3x$ to get $-2x$ . Multiply $(3x+1)$ by $-2x$ to get $-6x^2-2x$ . Subtract this from the previous remainder to get $12x-3$ .
Step $2$ : Divide $12x$ by $3x$ to get $4$ . Multiply $(3x+1)$ by $4$ to get $12x+4$ . Subtract this from the previous remainder to get $-7$ .
Step $3$ : So, the polynomial long division gives us $x^2 - 2x + 4 - \frac{7}{3x+1}$ .
Step $4$ : Now, integrate each term separately: $\int x^2 \, dx$ , $\int -2x \, dx$ , $\int 4 \, dx$ , and $\int -\frac{7}{3x+1} \, dx$ .
Step $5$ : The integral of $x^2$ is $(\frac{1}{3})x^3$ .
Step $6$ : The integral of $-2x$ is $-x^2$ .
Step $7$ : The integral of $4$ is $4x$ .
Step $8$ : The integral of $-\frac{7}{3x+1}$ is $-7\ln|3x+1|$ , because the derivative of $3x+1$ is $3$ , and we have to divide by that coefficient.
Step $9$ : So, the integral of the original function is $(\frac{1}{3})x^3 - x^2 + 4x - (\frac{7}{3})\ln|3x+1| + C$ .
Final Integration: Comparing with the answer choices, it looks like the correct answer is $(D)$ .