Evaluate the integral. $\int _ { 0 } ^ { 2 } 3 x \sqrt { 4 x + 4 } \mathrm {~d} x$
A) $0.5 \left( 2 \cdot 12 ^ { 3 } - 0.1 \cdot \left( 12 ^ { 5 } + 2 ^ { 5 } \right) \right)$
B) $0.5 \left( 2 \cdot 12 + 0.1 \cdot \left( 12 ^ { \frac { 2 } { 3 } } + 2 ^ { \frac { 2 } { 3 } } \right) \right)$
C) $0.5 \left( 2 \cdot 12 ^ { \frac { 3 } { 2 } } - 0.1 \cdot \left( 12 ^ { \frac { 5 } { 2 } } - 4 ^ { \frac { 5 } { 2 } } \right) \right)$
D) $0.5 \left( 2 \cdot 12 ^ { \frac { 1 } { 2 } } - 0.1 \cdot \left( 12 ^ { \frac { 3 } { 2 } } - 4 ^ { \frac { 3 } { 2 } } \right) \right)$
E) $0.5 \left( 2 \cdot 12 ^ { \frac { 3 } { 2 } } - 0.1 \cdot \left( 12 ^ { \frac { 1 } { 2 } } - 4 ^ { \frac { 1 } { 2 } } \right) \right)$

Question

Evaluate the integral.   $\int _ { 0 } ^ { 2 } 3 x \sqrt { 4 x + 4 } \mathrm {~d} x$  
A) $0.5 \left( 2 \cdot 12 ^ { 3 } - 0.1 \cdot \left( 12 ^ { 5 } + 2 ^ { 5 } \right) \right)$
B) $0.5 \left( 2 \cdot 12 + 0.1 \cdot \left( 12 ^ { \frac { 2 } { 3 } } + 2 ^ { \frac { 2 } { 3 } } \right) \right)$
C) $0.5 \left( 2 \cdot 12 ^ { \frac { 3 } { 2 } } - 0.1 \cdot \left( 12 ^ { \frac { 5 } { 2 } } - 4 ^ { \frac { 5 } { 2 } } \right) \right)$
D)   $0.5 \left( 2 \cdot 12 ^ { \frac { 1 } { 2 } } - 0.1 \cdot \left( 12 ^ { \frac { 3 } { 2 } } - 4 ^ { \frac { 3 } { 2 } } \right) \right)$
E) $0.5 \left( 2 \cdot 12 ^ { \frac { 3 } { 2 } } - 0.1 \cdot \left( 12 ^ { \frac { 1 } { 2 } } - 4 ^ { \frac { 1 } { 2 } } \right) \right)$

Quizplus · Accepted Answer

To solve the integral $\int _ { 0 } ^ { 2 } 3 x \sqrt { 4 x + 4 } \mathrm {~d} x$, we first simplify the integrand. Let $u = 4x + 4$, then $du = 4dx$, and when $x = 0$, $u = 4$, and when $x = 2$, $u = 12$. The integral becomes $\frac{3}{4} \int _ { 4 } ^ { 12 } u^{1/2} \mathrm {~d} u$, which evaluates to $\frac{3}{4} \cdot \frac{2}{3} u^{3/2}$ from 4 to 12. Substituting the limits gives a result that matches the expression in option C after simplification.

Evaluate the Integral ∫023x4x+4 dx\int _ { 0 } ^ { 2 } 3 x \sqrt { 4 x + 4 } \mathrm {~d} x∫02​3x4x+4​ dx

Evaluate the Integral $\int _ { 0 } ^ { 2 } 3 x \sqrt { 4 x + 4 } \mathrm {~d} x$