Evaluate the integral. $$\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x$$
A) $26.4$
B) $31.4$
C) $46.4$
D) $36.4$
E) $41.4$

Question

Quizplus · Accepted Answer

Explanation:Using the substitution $u = \sqrt{x}$, we have $du = \frac{1}{2\sqrt{x}} dx$ and $x = u^2$. Substituting, we get$$\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x = \int _ { 1 } ^ { 2 } \frac { u ^ { 4 } + 7 } { u } 2 u \ du = 2 \int _ { 1 } ^ { 2 } ( u ^ { 3 } + 7 ) d u$$$= 2 \left[ \frac { u ^ { 4 } } { 4 } + 7 u \right] _ { 1 } ^ { 2 } = 2 \left[ 4 + 14 - \frac { 1 } { 4 } - 7 \right] = 2 \left( \frac { 35 } { 4 } \right) = \frac { 35 } { 2 } = 17.5$

Evaluate the Integral ∫14x2+7xdx\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x∫14​x​x2+7​dx

Evaluate the Integral $\int _ { 1 } ^ { 4 } \frac { x ^ { 2 } + 7 } { \sqrt { x } } d x$