Indefinite integrals Compute the indefinite integral of the fol- lowing functions. r(t)=e3ti+11+t2j−12tk

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Step1 Let&#039;s recall the calculus of vector valued functions: We calculate the derivative of vector-valued functions by differentiating each component We obtain indefinite and definite integrals by computing antiderivatives of each component. Let&#039;s look at the theorem below in particular: Theorem. Let f→:R⇒R3 bea continuous vector-valued function f→(t)=⟨x(t),y(t),z(t)⟩, then the vector-valued integral is given by ∫f→(t)dt=⟨∫x(t)dt,∫y(t)dt,∫z(t)dt⟩. Step2 In order to integrate the given function we need to integrate each of the three components r(t)=e3ti+11+t2j−12tk ⇒∫r(t)dt=∫(e3tdt)i+∫(11+t2dt)j−∫(12tdt)k =(e3t3+C1)i+(tan−1⁡t+C2)j+(t+C3)k where C1,C2, and C3 are the constants of integration. Step3 Hence, the final answer: (e3t3+C1)i+(tan−1⁡t+C2)j+(t+C3)k

Compute the indefinite integral of the fol-lowing functions. r(t)=e^(3t)i+(1)/(1+t^2)j-(1)/(sqrt(2t))k

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