Math Problem Statement
Compute the integral 0 π⟨cos(2t),tet,t2⟩dt.
Solution
To compute the integral of the vector function from 0 to , we will integrate each component of the vector function separately over the interval .
The integral is:
= \langle \int_0^\pi \cos(2t) dt, \int_0^\pi te^t dt, \int_0^\pi t^2 dt \rangle$$ ### 1. First component: $$\int_0^\pi \cos(2t) dt$$ Using the standard integral of cosine: $$\int \cos(2t) dt = \frac{1}{2} \sin(2t)$$ Evaluating this from 0 to $$\pi$$: $$\left[ \frac{1}{2} \sin(2t) \right]_0^\pi = \frac{1}{2} (\sin(2\pi) - \sin(0)) = \frac{1}{2} (0 - 0) = 0$$ ### 2. Second component: $$\int_0^\pi te^t dt$$ This integral requires integration by parts. Let: $$u = t \quad \text{and} \quad dv = e^t dt$$ Then: $$du = dt \quad \text{and} \quad v = e^t$$ Using integration by parts: $$\int te^t dt = te^t - \int e^t dt = te^t - e^t$$ Evaluating from 0 to $$\pi$$: $$\left[ te^t - e^t \right]_0^\pi = \left( \pi e^\pi - e^\pi \right) - \left( 0e^0 - e^0 \right) = \pi e^\pi - e^\pi + 1$$ Thus, the second component is: $$(\pi - 1) e^\pi + 1$$ ### 3. Third component: $$\int_0^\pi t^2 dt$$ This is a standard power rule integral: $$\int t^2 dt = \frac{t^3}{3}$$ Evaluating from 0 to $$\pi$$: $$\left[ \frac{t^3}{3} \right]_0^\pi = \frac{\pi^3}{3} - 0 = \frac{\pi^3}{3}$$ ### Final Answer Combining the results from the three components, the integral of the vector function is: $$\int_0^\pi \langle \cos(2t), te^t, t^2 \rangle dt = \langle 0, (\pi - 1) e^\pi + 1, \frac{\pi^3}{3} \rangle$$ Would you like more details on any specific step or component? Here are 5 related questions you might find helpful: 1. How do you apply integration by parts to more complex functions? 2. What are the standard trigonometric integrals and their uses? 3. How can vector-valued functions be integrated in higher dimensions? 4. What are some applications of integrating vector functions? 5. How do boundary conditions affect the results of definite integrals? **Tip:** Remember that for vector integrals, integrating each component separately simplifies the computation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Vector Functions
Integration
Formulas
∫ cos(2t) dt = (1/2) sin(2t)
∫ te^t dt = te^t - e^t
∫ t^2 dt = (t^3)/3
Theorems
Integration by Parts
Power Rule for Integration
Trigonometric Integral
Suitable Grade Level
Undergraduate Level (Calculus II)
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