Evaluate the integral.
-$$\int _ { 0 } ^ { 4 } \left[ - \mathbf { i } - \frac { 2 t } { \left( 7 + t ^ { 2 } \right) ^ { 2 } } j + \frac { 15 } { 2 } \sqrt { t k } \right] d t$$
A) $+ 4 i + \frac { 16 } { 161 } j + 40 k$
B) $- 4 i - \frac { 16 } { 161 } j - 40 k$
C) $- 4 i - \frac { 16 } { 161 } j + 40 k$
D) $- 4 i + \frac { 4 } { 77 } j + 40 k$

Question

Quizplus · Accepted Answer

The integral can be evaluated component-wise. For the $i$ component, the integral of a constant $-i$ over the interval from 0 to 4 is simply $-4i$. For the $j$ component, the integral involves a rational function, which, when integrated from 0 to 4, results in $-\frac{16}{161}j$. For the $k$ component, integrating $\frac{15}{2}\sqrt{t}$ from 0 to 4 yields $40k$. Therefore, the correct answer is $- 4 i - \frac { 16 } { 161 } j + 40 k$.

Evaluate the Integral ∫04[−i−2t(7+t2)2j+152tk]dt\int _ { 0 } ^ { 4 } \left[ - \mathbf { i } - \frac { 2 t } { \left( 7 + t ^ { 2 } \right) ^ { 2 } } j + \frac { 15 } { 2 } \sqrt { t k } \right] d t∫04​[−i−(7+t2)22t​j+215​tk​]dt

Evaluate the Integral $\int _ { 0 } ^ { 4 } \left[ - \mathbf { i } - \frac { 2 t } { \left( 7 + t ^ { 2 } \right) ^ { 2 } } j + \frac { 15 } { 2 } \sqrt { t k } \right] d t$