On this page we explain how to evaluate integrals of vector functions.
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Integrals of vector functions use special techniques. Here is a video clip that should help you a lot.
video by Dr Chris Tisdell 

To put this all together, here is a full lecture on derivatives and integrals of vector functions.
video by Prof Leonard 

Try these practice problems.
Practice
Unless otherwise instructed, evaluate these integrals.
\(\displaystyle{ \int_{0}^{\pi/2}{ \vec{r}(t)~dt } }\) for \( \vec{r}(t) = [3\sin^2t\cos t]\,\hat{i}+ \) \( [3\sin t\cos^2t]\,\hat{j} + \) \( [2\sin t\cos t]\,\hat{k} \)
Problem Statement 

Evaluate \(\displaystyle{ \int_{0}^{\pi/2}{ \vec{r}(t)~dt } }\) for \( \vec{r}(t) = [3\sin^2t\cos t]\,\hat{i}+ \) \( [3\sin t\cos^2t]\,\hat{j} + \) \( [2\sin t\cos t]\,\hat{k} \)
Final Answer 

\( \hat{i} + \hat{j} + \hat{k} \)
Problem Statement
Evaluate \(\displaystyle{ \int_{0}^{\pi/2}{ \vec{r}(t)~dt } }\) for \( \vec{r}(t) = [3\sin^2t\cos t]\,\hat{i}+ \) \( [3\sin t\cos^2t]\,\hat{j} + \) \( [2\sin t\cos t]\,\hat{k} \)
Solution
video by Krista King Math 

Final Answer
\( \hat{i} + \hat{j} + \hat{k} \)
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\(\displaystyle{ \int{ \frac{5}{t^2}\vhat{i}  4\sqrt{t}\vhat{j} ~dt } }\)
Problem Statement 

Evaluate \(\displaystyle{ \int{ \frac{5}{t^2}\vhat{i}  4\sqrt{t}\vhat{j} ~dt } }\)
Final Answer 

\(\displaystyle{ \left[ \frac{5}{t}+c_1\right] \vhat{i}  \left[ \frac{8}{3}t^{3/2} + c_2\right] \vhat{j} }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{5}{t^2}\vhat{i}  4\sqrt{t}\vhat{j} ~dt } }\)
Solution
video by MIP4U 

Final Answer
\(\displaystyle{ \left[ \frac{5}{t}+c_1\right] \vhat{i}  \left[ \frac{8}{3}t^{3/2} + c_2\right] \vhat{j} }\)
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\(\displaystyle{ \int{ \frac{2}{t}\vhat{i}  \sin(t)\vhat{j} + \sec^2(2t)\vhat{k} ~dt } }\)
Problem Statement 

Evaluate \(\displaystyle{ \int{ \frac{2}{t}\vhat{i}  \sin(t)\vhat{j} + \sec^2(2t)\vhat{k} ~dt } }\)
Final Answer 

\(\displaystyle{ \left[ 2\ln(t)+c_1\right]\vhat{i} + \left[ \cos(t)+c_2 \right]\vhat{j} + \left[ (1/2)\tan(2t) + c_3\right]\vhat{k} }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{2}{t}\vhat{i}  \sin(t)\vhat{j} + \sec^2(2t)\vhat{k} ~dt } }\)
Solution
video by MIP4U 

Final Answer
\(\displaystyle{ \left[ 2\ln(t)+c_1\right]\vhat{i} + \left[ \cos(t)+c_2 \right]\vhat{j} + \left[ (1/2)\tan(2t) + c_3\right]\vhat{k} }\)
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Practice Instructions
Unless otherwise instructed, evaluate these integrals.