Evaluate int(2x^(3)+7x^(2)+2x+9)/(2x+3)dx. Choose 1 answer: (A) (x^(3))/(3)+x^(2)-2x+(15 ln |2x+3|)/(2)+C (B) (x^(3))/(3)+x^(2)-2x+(5ln |2x+3|)/(2)+C (C) (x^(3))/(3)+(x^(2))/(2)-2x+(15 ln |2x+3|)/(2)+C (D) (x^(3))/(3)+(x^(2))/(2)-2x+(5ln |2x+3|)/(2)+C

Divide and Multiply: Divide 2x^3 by 2x to get x^2. Multiply (2x + 3) by x^2 to get 2x^3 + 3x^2. Subtract this from the original polynomial to get 4x^2 + 2x + 9.Subtract and Simplify: Now divide 4x^2 by 2x to get 2x. Multiply (2x + 3) by 2x to get 4x^2 + 6x. Subtract this from the remaining polynomial to get -4x + 9.Divide and Multiply: Divide -4x by 2x to get -2. Multiply (2x + 3) by -2 to get -4x - 6. Subtract this from the remaining polynomial to get 15.Subtract and Simplify: So, the polynomial long division gives us x^2 + 2x - 2 + 15/(2x + 3). Now we can integrate each term separately.Integrate Each Term: Integrate x^2 to get (1/3)x^3. Integrate 2x to get x^2. Integrate -2 to get -2x. Integrate 15/(2x + 3) to get (15/2)ln|2x + 3|. Don't forget the constant of integration C.Combine Integrated Parts: Combine all the integrated parts to get the final answer: (1/3)x^3 + x^2 - 2x + (15/2)ln|2x + 3| + C.

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Evaluate
int(2x^(3)+7x^(2)+2x+9)/(2x+3)dx.
Choose 1 answer:
(A)
(x^(3))/(3)+x^(2)-2x+(15 ln |2x+3|)/(2)+C
(B)
(x^(3))/(3)+x^(2)-2x+(5ln |2x+3|)/(2)+C
(C)
(x^(3))/(3)+(x^(2))/(2)-2x+(15 ln |2x+3|)/(2)+C
(D)
(x^(3))/(3)+(x^(2))/(2)-2x+(5ln |2x+3|)/(2)+C

Evaluate $\int \frac{2 x^{3}+7 x^{2}+2 x+9}{2 x+3} d x$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{x^{3}}{3}+x^{2}-2 x+\frac{15 \ln |2 x+3|}{2}+C$ $\newline$ (B) $\frac{x^{3}}{3}+x^{2}-2 x+\frac{5 \ln |2 x+3|}{2}+C$ $\newline$ (C) $\frac{x^{3}}{3}+\frac{x^{2}}{2}-2 x+\frac{15 \ln |2 x+3|}{2}+C$ $\newline$ (D) $\frac{x^{3}}{3}+\frac{x^{2}}{2}-2 x+\frac{5 \ln |2 x+3|}{2}+C$

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Math Problems

Calculus

Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate

\int \frac{2 x^{3}+7 x^{2}+2 x+9}{2 x+3} d x

\newline

Choose

1

answer:

\newline

(A)

\frac{x^{3}}{3}+x^{2}-2 x+\frac{15 \ln |2 x+3|}{2}+C

\newline

(B)

\frac{x^{3}}{3}+x^{2}-2 x+\frac{5 \ln |2 x+3|}{2}+C

\newline

(C)

\frac{x^{3}}{3}+\frac{x^{2}}{2}-2 x+\frac{15 \ln |2 x+3|}{2}+C

\newline

(D)

\frac{x^{3}}{3}+\frac{x^{2}}{2}-2 x+\frac{5 \ln |2 x+3|}{2}+C

Divide and Multiply: Divide $2x^3$ by $2x$ to get $x^2$ . Multiply $(2x + 3)$ by $x^2$ to get $2x^3 + 3x^2$ . Subtract this from the original polynomial to get $4x^2 + 2x + 9$ .
Subtract and Simplify: Now divide $4x^2$ by $2x$ to get $2x$ . Multiply $(2x + 3)$ by $2x$ to get $4x^2 + 6x$ . Subtract this from the remaining polynomial to get $-4x + 9$ .
Divide and Multiply: Divide $-4x$ by $2x$ to get $-2$ . Multiply $(2x + 3)$ by $-2$ to get $-4x - 6$ . Subtract this from the remaining polynomial to get $15$ .
Subtract and Simplify: So, the polynomial long division gives us $x^2 + 2x - 2 + \frac{15}{2x + 3}$ . Now we can integrate each term separately.
Integrate Each Term: Integrate $x^2$ to get $(1/3)x^3$ . Integrate $2x$ to get $x^2$ . Integrate $-2$ to get $-2x$ . Integrate $15/(2x + 3)$ to get $(15/2)\ln|2x + 3|$ . Don't forget the constant of integration $C$ .
Combine Integrated Parts: Combine all the integrated parts to get the final answer: $(\frac{1}{3})x^3 + x^2 - 2x + (\frac{15}{2})\ln|2x + 3| + C$ .