Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$
A) $\frac { \left( x ^ { 2 } - 3 \right) ^ { 8 } x ^ { 3 } } { 24 } + C$
B) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 144 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$
C) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 72 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$
D) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } + C$

Question

Quizplus · Accepted Answer

To solve the integral $\int x^3 (x^2 - 3)^7 dx$, we can use substitution. Let $u = x^2 - 3$, then $du = 2x dx$. However, since we have $x^3$ and not $x$, we adjust to $x^2 dx = \frac{1}{2} du$. The integral becomes $\frac{1}{2} \int u^7 du$, which is not directly applicable due to the $x^3$ term. The correct approach involves integrating by parts or recognizing the need to express $x^3$ in terms of $u$ and $x$, which complicates direct substitution. The correct answer involves recognizing the integral can be simplified and solved through a combination of substitution and algebraic manipulation, leading to the expression in choice B, which correctly accounts for the integration process and the algebraic manipulation needed to express the original integral in terms of $u$ and then back in terms of $x$.

Evaluate ∫x3(x2−3)7dx\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x∫x3(x2−3)7dx

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$