Evaluate the integral $\int \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } x ^ { 4 } d x$
A) $\frac { 1 } { 140 } \left( 4 x ^ { 5 } + 3 \right) ^ { 7 } + C$
B) $\frac { 20 } { 9 } \left( 4 x ^ { 5 } + 3 \right) ^ { 9 } + C$
C) $\frac { 5 } { 2 } \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } + C$
D) $\frac { 1 } { 180 } \left( 4 x ^ { 5 } + 3 \right) ^ { 9 } + C$
E) $\frac { 1 } { 160 } \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } + C$

Question

Quizplus · Accepted Answer

To solve the integral $\int \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } x ^ { 4 } d x$, we use substitution. Let $u = 4x^5 + 3$, then $du = 20x^4 dx$. The integral becomes $\int u^8 \frac{1}{20} du$, which simplifies to $\frac{1}{20} \int u^8 du$. Integrating $u^8$ gives $\frac{u^9}{9}$, so the integral is $\frac{1}{20} \cdot \frac{u^9}{9} + C = \frac{1}{180} (4x^5 + 3)^9 + C$.

Evaluate the Integral ∫(4x5+3)8x4dx\int \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } x ^ { 4 } d x∫(4x5+3)8x4dx

Evaluate the Integral $\int \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } x ^ { 4 } d x$