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

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

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Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate

\int \frac{x^{3}+x}{x+1} d x

.

\newline

Choose

1

answer:

\newline

(A)

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

\newline

(B)

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

\newline

(C)

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

\newline

(D)

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

Perform Long Division: Long division of $x^3 + x$ by $x + 1$ to simplify the integrand. $\newline$ $\frac{x^3 + x}{x + 1} = x^2 - x + 2 - \frac{2}{x + 1}$
Integrate Each Term: Integrate each term separately. $\int x^2 \, dx = \frac{x^3}{3}$ , $\int (-x) \, dx = -\frac{x^2}{2}$ , $\int 2 \, dx = 2x$ , \int \left(-\frac{\(2\)}{x + \(1\)}\right) dx = \(-2\ln|x + $1$ |
Combine Integration Results: Combine the results of the integrations. $\frac{x^3}{3} - \frac{x^2}{2} + 2x - 2\ln|x + 1| + C$

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