91Ó°ÊÓ

Evaluate the following integrals:\(\int\left(e^{t} \mathbf{i}+\sin t \mathbf{j}+\frac{1}{2 t-1} \mathbf{k}\right) d t\)

Find the derivatives of u(t), u?(t), u?(t)×u(t), u(t)×u?(t), and u(t)?u?(t). Find the unit tangent vector. $$ \mathbf{u}(t)=\left\langle e^{t}, e^{-t}\right\rangle $$

Evaluate the following integrals:\(\int_{0}^{1} \mathbf{r}(t) d t\), where \(\mathbf{r}(t)=\left\langle\sqrt[3]{t}, \frac{1}{t+1}, e^{-t}\right\rangle\)

Find the derivatives of u(t), u?(t), u?(t)×u(t), u(t)×u?(t), and u(t)?u?(t). Find the unit tangent vector. $$ \mathbf{u}(t)=\left\langle t^{2}, 2 t+6,4 t^{5}-12\right\rangle $$

Evaluate the following integrals. $$ \int\left(\tan (t) \sec (t) \mathbf{i}-t e^{3 t} \mathbf{j}\right) d t $$

Evaluate the following integrals. $$ \int_{1}^{4} \mathbf{u}(t) d t, \text { with } \mathbf{u}(t)=\left\langle\frac{\ln (t)}{t}, \frac{1}{\sqrt{t}}, \sin \left(\frac{t \pi}{4}\right)\right\rangle $$

Find the length for the following curves. $$ \mathbf{r}(t)=\langle 3(t), 4 \cos (t), 4 \sin (t)\rangle \text { for } 1 \leq t \leq 4 $$

Find the length for the following curves. $$ \mathbf{r}(t)=2 \mathbf{i}+\mathbf{t} \mathbf{j}+3 t^{2} \mathbf{k} \text { for } 0 \leq t \leq 1 $$

Reparameterize the following functions with respect to their arc length measured from t=0 in direction of increasing t. $$ \mathbf{r}(t)=2 t \mathbf{i}+(4 t-5) \mathbf{j}+(1-3 t) \mathbf{k} $$

Reparameterize the following functions with respect to their arc length measured from t=0 in direction of increasing t. $$ \mathbf{r}(t)=\cos (2 t) \mathbf{i}+8 t \mathbf{j}-\sin (2 t) \mathbf{k} $$