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17Calculus - Integrals of Vector Functions

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

On this page we explain how to evaluate integrals of vector functions.

Integrals of vector functions use special techniques. Here is a video clip that should help you a lot.

Dr Chris Tisdell - integrals of vector functions [6mins-57secs]

video by Dr Chris Tisdell

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

Prof Leonard - Calculus 3 Lecture 12.2: Derivatives and Integrals of Vector Functions [2hrs-42mins-18secs]

video by Prof Leonard

Try these practice problems.

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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

Krista King Math - 2026 video 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

MIP4U - 2037 video 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

MIP4U - 2038 video 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.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

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