Pregunta
Use a change of variables or the table to evaluate the following indefinite integral. \( \int x^{3} e^{x^{4}} d x \) \( \int x^{3} e^{x^{4}} d x=\square \)
Ask by Weston Cox. in the United States
Mar 07,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\[
\int x^{3} e^{x^{4}}\, dx = \frac{1}{4} e^{x^{4}} + C
\]
Solución
Let \( u = x^4 \). Then the derivative is
\[
\frac{du}{dx} = 4x^3,
\]
so that
\[
x^3\, dx = \frac{1}{4}\, du.
\]
Substitute into the integral:
\[
\int x^3 e^{x^4}\, dx = \int e^u \cdot \frac{1}{4}\, du = \frac{1}{4} \int e^u\, du.
\]
Evaluating the integral, we have:
\[
\int e^u\, du = e^u.
\]
Thus,
\[
\int x^3 e^{x^4}\, dx = \frac{1}{4} e^u + C.
\]
Substitute back \( u = x^4 \) to obtain the final answer:
\[
\int x^3 e^{x^4}\, dx = \frac{1}{4} e^{x^4} + C.
\]
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Simplifique esta solución Mind Expander
To solve the integral \( \int x^{3} e^{x^{4}} d x \), we can use a change of variables. Let's set \( u = x^4 \). Then, the derivative \( du = 4x^3 dx \), or \( dx = \frac{du}{4x^3} \). The integral becomes: \[ \int x^{3} e^{x^{4}} d x = \int x^{3} e^{u} \frac{du}{4x^{3}} = \frac{1}{4} \int e^{u} du \] Now, integrating \( e^{u} \) gives us \( e^{u} + C \). Re-substituting \( u = x^4 \): \[ \frac{1}{4}(e^{x^{4}} + C) = \frac{1}{4} e^{x^{4}} + C \] So, the final result is: \[ \int x^{3} e^{x^{4}} d x = \frac{1}{4} e^{x^{4}} + C \]
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