Evaluate the integral.
-$\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 5 + 3 x ^ { 3 } \right) ^ { 2 } }$ dx
A) $\frac { 43 } { 3 }$
B) $\frac { 1 } { 120 }$
C) $\frac { 13 } { 6 }$
D) $\frac { 27 } { 40 }$

Question

Evaluate the integral. 
-$\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 5 + 3 x ^ { 3 } \right) ^ { 2 } }$ dx
A) $\frac { 43 } { 3 }$
B) $\frac { 1 } { 120 }$
C) $\frac { 13 } { 6 }$
D) $\frac { 27 } { 40 }$

Quizplus · Accepted Answer

To solve the integral $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 5 + 3 x ^ { 3 } \right) ^ { 2 } }$ dx, we can use a substitution method. Let $u = 5 + 3x^3$, then $du = 9x^2 dx$. However, to match the integral's $x^2 dx$ part, we adjust to $du/9 = x^2 dx$. The bounds of integration change with the substitution, at $x = 0$, $u = 5$, and at $x = 1$, $u = 8$. Thus, the integral becomes $\int _ { 5 } ^ { 8 } \frac { 1 } { 9u^2 }$ du. This integral is straightforward to evaluate and leads to the result $\frac { 1 } { 120 }$, making option B correct.

Evaluate the Integral ∫01x2(5+3x3)2\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 5 + 3 x ^ { 3 } \right) ^ { 2 } }∫01​(5+3x3)2x2​

Evaluate the Integral $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 5 + 3 x ^ { 3 } \right) ^ { 2 } }$