Question

(a) \[ \int\left[e^{t} \vec{i}+2 t \vec{j}+\ln t \vec{k}\right] d t \] (b) \[ \int_{1}^{2}\left[t^{2} \vec{i}+(t \sqrt{t-1}) \vec{j}+(t \sin \pi t) \vec{k}\right] d t \] (c) \[ \int_{0}^{1}\left[\left(\frac{4}{1+t^{2}}\right) \vec{j}+\left(\frac{2 t}{1+t^{2}}\right) \vec{k}\right] d t \] (d) \[ \int_{0}^{\frac{\pi}{2}}\left[\left(3 \sin ^{2} t \cos t\right) \vec{i}+\left(3 \sin t \cos ^{2} t\right) \vec{j}+(2 \sin t \cos t) \vec{k}\right] d t \]

          (a)
\[
\int\left[e^{t} \vec{i}+2 t \vec{j}+\ln t \vec{k}\right] d t
\]
(b)
\[
\int_{1}^{2}\left[t^{2} \vec{i}+(t \sqrt{t-1}) \vec{j}+(t \sin \pi t) \vec{k}\right] d t
\]
(c)
\[
\int_{0}^{1}\left[\left(\frac{4}{1+t^{2}}\right) \vec{j}+\left(\frac{2 t}{1+t^{2}}\right) \vec{k}\right] d t
\]
(d)
\[
\int_{0}^{\frac{\pi}{2}}\left[\left(3 \sin ^{2} t \cos t\right) \vec{i}+\left(3 \sin t \cos ^{2} t\right) \vec{j}+(2 \sin t \cos t) \vec{k}\right] d t
\]

(a)

∫[e^ti⃗+2 t j⃗+ln t k⃗] d t

(b)

∫1^2[t^2i⃗+(t √(t-1)) j⃗+(t sinπ t) k⃗] d t

(c)

∫0^1[((4)/(1+t^2)) j⃗+((2 t)/(1+t^2)) k⃗] d t

(d)

∫0^(π)/(2)[(3 sin ^2 t cos t) i⃗+(3 sin t cos ^2 t) j⃗+(2 sin t cos t) k⃗] d t

Added by Jeremy W.

Thomas Calculus

George B. Thomas Jr. 14th Edition

Chapter 16

Instant Answer

Solved by Expert Adi S

07/05/2024

Step 1

The integral of \(e^{t}\) is \(e^{t}\), the integral of \(2t\) is \(t^{2}\), and the integral of \(\ln t\) is \(t \ln t - t\). So the result is: \[e^{t} \vec{i} + t^{2} \vec{j} + (t \ln t - t) \vec{k} + \vec{C}\] where \(\vec{C}\) is the constant of integration, a Show more…

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