Evaluate the integral.
-$\int _ { 0 } ^ { 2 } \frac { 12 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$ dx
A) $\frac { 728 } { 19,683 }$
B) $\frac { 8 } { 3 }$
C) $\frac { 8 } { 27 }$
D) 2

Question

Evaluate the integral. 
-$\int _ { 0 } ^ { 2 } \frac { 12 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$ dx
A) $\frac { 728 } { 19,683 }$
B) $\frac { 8 } { 3 }$
C) $\frac { 8 } { 27 }$
D) 2

Quizplus · Accepted Answer

The integral can be solved by making a substitution. Let $u = 2x^2 + 1$, then $du = 4x dx$. Adjusting for the 12 in the numerator, we rewrite the integral in terms of $u$, which simplifies to a form that can be directly integrated. After performing the integration and substituting back for $x$, evaluating the definite integral from 0 to 2 gives $\frac{8}{3}$.

Evaluate the Integral ∫0212x(2x2+1)2\int _ { 0 } ^ { 2 } \frac { 12 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }∫02​(2x2+1)212x​

Evaluate the Integral $\int _ { 0 } ^ { 2 } \frac { 12 x } { \left( 2 x ^ { 2 } + 1 \right) ^ { 2 } }$