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) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } + C$
C) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 144 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$
D) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 72 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$

Question

Quizplus · Accepted Answer

This integral can be solved using the substitution method. Let $u = x^2 - 3$, then $du = 2x dx$. However, the integral has $x^3$, so we rewrite $x^3$ as $x \cdot x^2 = x(u + 3)$. Substituting $u$ and $du$ into the integral and integrating, we get a polynomial in $u$ which, when converted back to $x$, matches option C. This option correctly accounts for the integration of $u^7$ multiplied by $x$, and the subtraction of a term to adjust for the initial substitution, leading to the correct antiderivative.

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

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