Weston Knight
08/24/2023 · Primary School
Encontrar \( \int\left[\frac{t}{t^{4}+10} \mathbf{i}+\frac{1-\cos 3 t}{4} d t \mathbf{j}\right] d t \)
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Para resolver la integral vectorial
\[
\int\left[\frac{t}{t^{4}+10} \mathbf{i}+\frac{1-\cos 3 t}{4} \mathbf{j}\right] d t,
\]
integra cada componente por separado. La componente en \(\mathbf{i}\) no tiene una forma elemental simple, mientras que la componente en \(\mathbf{j}\) se integra fácilmente a \( \frac{t}{4} - \frac{1}{12} \sin 3t \). La integral vectorial final es
\[
\left( \int \frac{t}{t^{4}+10} dt \right) \mathbf{i} + \left( \frac{t}{4} - \frac{1}{12} \sin 3t \right) \mathbf{j} + \mathbf{C},
\]
donde \(\mathbf{C}\) es una constante vectorial.
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