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

Polynomial Long Division: First, let's try polynomial long division to simplify the integrand. Divide 2x^3 by x to get 2x^2. Multiply (x-2) by 2x^2 to get 2x^3 - 4x^2. Subtract this from the original polynomial to get x^2 - 3x + 2.Simplify Remainder: Now, divide x^2 by x to get x. Multiply (x-2) by x to get x^2 - 2x. Subtract this from the remainder to get -x + 2.Integrate Quotient: Finally, divide -x by x to get -1. Multiply (x-2) by -1 to get -x + 2. Subtract this from the remainder to get 0. So the quotient is 2x^2 + x - 1 and the remainder is 0.Final Integration: Now we integrate the quotient 2x^2 + x - 1 term by term. The integral of 2x^2 is (2/3)x^3, the integral of x is (1/2)x^2, and the integral of -1 is -x. Don't forget the constant of integration C.So the integral of (2x^(3)-3x^(2)-3x+2)/(x-2)dx is (2/3)x^3 + (1/2)x^2 - x + C.

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

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

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Find derivatives of using multiple formulae

Full solution

Q. Evaluate

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

\newline

Choose

1

answer:

\newline

(A)

\frac{2 x^{3}}{3}+\frac{x^{2}}{2}-x+C

\newline

(B)

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

\newline

(C)

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

\newline

(D)

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

Polynomial Long Division: First, let's try polynomial long division to simplify the integrand. $\newline$ Divide $2x^3$ by $x$ to get $2x^2$ . Multiply $(x-2)$ by $2x^2$ to get $2x^3 - 4x^2$ . Subtract this from the original polynomial to get $x^2 - 3x + 2$ .
Simplify Remainder: Now, divide $x^2$ by $x$ to get $x$ . Multiply $(x-2)$ by $x$ to get $x^2 - 2x$ . Subtract this from the remainder to get $-x + 2$ .
Integrate Quotient: Finally, divide $-x$ by $x$ to get $-1$ . Multiply $(x-2)$ by $-1$ to get $-x + 2$ . Subtract this from the remainder to get $0$ . So the quotient is $2x^2 + x - 1$ and the remainder is $0$ .
Final Integration: Now we integrate the quotient $2x^2 + x - 1$ term by term. $\newline$ The integral of $2x^2$ is $(\frac{2}{3})x^3$ , the integral of $x$ is $(\frac{1}{2})x^2$ , and the integral of $-1$ is $-x$ . Don't forget the constant of integration $C$ .
Final Integration: Now we integrate the quotient $2x^2 + x - 1$ term by term. $\newline$ The integral of $2x^2$ is $(2/3)x^3$ , the integral of $x$ is $(1/2)x^2$ , and the integral of $-1$ is $-x$ . Don't forget the constant of integration $C$ .So the integral of $(2x^{3}-3x^{2}-3x+2)/(x-2)dx$ is $($2$/$3$)x^$3$ + ($1$/$2$)x^$2$ - x + C.