In this exercise, compute the indefinite integral of the following function.

r(t)=$2^t i+\dfrac{1}{1+2 t} j+\ln t k$

In this exercise, compute the indefinite integral of the following function.

r(t)=$t e^t \mathbf{i}+t \sin t^2 \mathbf{j}-\dfrac{2 t}{\sqrt{t^2+4}} \mathbf{k}$

In this exercise, compute the indefinite integral of the following function.

r(t)=$\langle 2 \cos t, 2 \sin 3 t, 4 \cos 8 t\rangle$

In this exercise, compute the indefinite integral of the following function.

r(t)=$\left\langle 5 t^{-4}-t^2, t^6-4 t^3, \dfrac{2}{t}\right\rangle$

find the solution.x(x2+y2)3/2+y(x2+y2)3/2dydx=0

In this exercise, how do you determine whether r(t)=f(t)i+g(t)j+h(t)k is continuous at t=a?

In this exercise, how do you evaluate the given integral $\displaystyle\int_a^b \mathbf{r}(t) d t$ ?

Evaluate the line integral, where C is the given curve. $\int_{C} \sin x d x+\cos y d y, \quad C$ consists of the top half of the circle $x^{2}+y^{2}=1$ from (1, 0) to (-1, 0) and the line segment from (-1, 0) to (-2, 3)

Compute the line integral of the scalar function over the curve.

f(x, y, z)=3zy-1+12xz, $\mathbf{c}(t)=\left(t, \dfrac{t^2}{\sqrt{2}}, \dfrac{t^3}{3}\right), \quad 0 \leq t \leq 2$

Compute the line integral of the vector field over the oriented curve.

$\mathbf{F}=\left\langle x^2, x y\right\rangle, \quad$ line segment from $(0,0)$ to $(2,2)$

Compute the line integral of the vector field over the oriented curve.

$\mathbf{F}=\langle y+z, x, y\rangle, \quad \mathbf{c}(t)=\left(t+t^2, \dfrac{1}{3} t^3, 2+t\right)$ for $0 \leq t \leq 2$

Evaluate the line integral, where C is the given curve. integral C xy^4 ds, C is the right half of the circle x^2+y^2=16

Given the velocity of an object and its initial position, how do you find the position of the object, for $t \geq 0$?

In this exercise, compute the indefinite integral of the following function.

r(t)=$\left\langle t^4-3 t, 2 t-1,10\right\rangle$