Use a table of integrals to evaluate the integral. $\int x \sqrt { 2 + 2 x } d x$
A) $\frac { 2 \sqrt { 2 } } { 15 } ( 2 x - 1 ) ( x + 1 ) ^ { 3 / 2 } + C$
B) $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) ( x + 1 ) ^ { 3 / 2 } + C$
C) $\frac { 2 } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$
D) $\frac { 2 \sqrt { 2 } } { 15 } ( 3 x - 2 ) \sqrt { x + 1 } + C$

Question

Quizplus · Accepted Answer

To solve the integral $\int x \sqrt { 2 + 2 x } d x$, we first simplify the expression under the square root to get $\sqrt{2(1+x)}$. Using a substitution method or referring to a table of integrals for the form $\int x(a+bx)^{n}dx$, we find that the correct integral involves a polynomial expression times $(x+1)^{3/2}$, which matches option B. The coefficients and powers in option B correctly follow from the integration process, including the adjustment for the $2\sqrt{2}$ factor that comes from the $2+2x$ under the square root and the distribution of powers and coefficients in the result.

Use a Table of Integrals to Evaluate the Integral ∫x2+2xdx\int x \sqrt { 2 + 2 x } d x∫x2+2x​dx

Use a Table of Integrals to Evaluate the Integral $\int x \sqrt { 2 + 2 x } d x$