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 $\int x \sqrt { 2 + 2 x } d x$, first simplify the expression under the square root to get $\sqrt{2(1+x)}$, which can be further simplified to $\sqrt{2}\sqrt{1+x}$. Then, use a substitution if necessary or directly apply a formula from the table of integrals for the form $\int x(a+bx)^{n} dx$, which typically results in a combination of $x$ and $(a+bx)^{n+1}$ terms, multiplied by constants. The correct answer must correctly apply these principles, leading to option B as it correctly integrates the given function and matches the form expected from such an integral, involving $(x+1)^{3/2}$ and a linear term in $x$, all scaled by constants.

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 $\int x \sqrt { 2 + 2 x } d x$