Evaluate $\int x ( x - 2 ) ^ { 9 } d x$
A) $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( 5 x + 1 ) + C$
B) $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 10 x + 1 ) + C$
C) $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 5 x + 2 ) + C$
D) $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( x + 10 ) + C$

Question

Quizplus · Accepted Answer

To solve the integral $\int x ( x - 2 ) ^ { 9 } d x$, we can use integration by parts or substitution. Let's use substitution for simplicity. Let $u = x - 2$, then $du = dx$, and when $x = u + 2$, the integral becomes $\int (u + 2)u^9 du$. Expanding this gives $\int (u^{10} + 2u^9) du$, which integrates to $\frac{1}{11}u^{11} + \frac{1}{5}u^{10} + C$. Substituting back $u = x - 2$, we get $\frac{1}{11}(x - 2)^{11} + \frac{1}{5}(x - 2)^{10} + C$. Simplifying and combining terms to match the answer format, we get $\frac{1}{55}(x - 2)^{10}(5x + 1) + C$, which corresponds to option A.

Evaluate ∫x(x−2)9dx\int x ( x - 2 ) ^ { 9 } d x∫x(x−2)9dx

Evaluate $\int x ( x - 2 ) ^ { 9 } d x$