Evaluate the integral.
-$$\int _ { 0 } ^ { 1 } \left[ \frac { 1 } { \sqrt { 2 t - t ^ { 2 } } } i + t e ^ { 4 } t j - \frac { 15 t ^ { 2 } } { \left( 4 + 5 t ^ { 3 } \right) ^ { 2 } } \mathbf { k } \right] d t$$
A) $\frac { 3 e ^ { 4 } + 1 } { 16 } \mathbf { j } + \frac { 5 } { 24 } \mathbf { k }$
B) $\frac { \pi } { 2 } \mathrm { i } + \frac { 3 \mathrm { e } ^ { 4 } + 1 } { 16 } \mathrm { j } - \frac { 5 } { 36 } \mathbf { k }$
C) $- \frac { 3 } { 2 } \pi \mathbf { i } + \frac { 3 \mathrm { e } ^ { 4 } + 1 } { 16 } \mathbf { j } + \frac { 5 } { 4 } \mathbf { k }$
D) $- \frac { 3 } { 2 } \pi \mathbf { i } + \frac { 3 \mathrm { e } ^ { 4 } } { 16 } \mathbf { j } + \frac { 5 } { 24 } \mathbf { k }$

Question

Quizplus · Accepted Answer

The integral can be evaluated by integrating each component separately. For the i component, the integral of $1/\sqrt{2t-t^2}$ from 0 to 1 is $\pi/2$ because it represents a semicircle with radius 1/2. For the j component, integrating $te^{4t}$ from 0 to 1 gives $\frac{3e^4+1}{16}$. For the k component, integrating $-\frac{15t^2}{(4+5t^3)^2}$ from 0 to 1 gives $-\frac{5}{36}$.

Evaluate the Integral A) 3e4+116j+524k\frac { 3 e ^ { 4 } + 1 } { 16 } \mathbf { j } + \frac { 5 } { 24 } \mathbf { k }163e4+1​j+245​k

Evaluate the Integral A) $\frac { 3 e ^ { 4 } + 1 } { 16 } \mathbf { j } + \frac { 5 } { 24 } \mathbf { k }$