Question
Calcula \( \frac{d}{d x}\left[\int_{0}^{X} \sqrt{t^{6}+4 d t}\right] \)
Ask by Griffiths Greene. in Mexico
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
La derivada es \( \sqrt{X^{6} + 4 d X} \).
Solution
La derivada de la integral
\[
\frac{d}{d x}\left[\int_{0}^{X} \sqrt{t^{6}+4 d t}\, dt\right]
\]
se puede calcular aplicando el **Teorema Fundamental del Cálculo**. Este teorema establece que si \( F(X) = \int_{a}^{X} f(t)\, dt \), entonces
\[
F'(X) = f(X)
\]
Aplicando esto al caso dado:
\[
\frac{d}{d X}\left[\int_{0}^{X} \sqrt{t^{6}+4 d t}\, dt\right] = \sqrt{X^{6} + 4 d X}
\]
Por lo tanto, la derivada es:
\[
\sqrt{X^{6} + 4 d X}
\]
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Simplify this solution The Deep Dive
To differentiate the integral \( \frac{d}{d x}\left[\int_{0}^{X} \sqrt{t^{6}+4} \, dt\right] \), we can use the Fundamental Theorem of Calculus. This theorem states that if \( F(X) = \int_{a}^{X} f(t) \, dt \), then \( \frac{dF}{dX} = f(X) \). Here, let \( f(t) = \sqrt{t^{6} + 4} \). Therefore, we have: \[ \frac{d}{dx}\left[\int_{0}^{X} \sqrt{t^{6}+4} \, dt\right] = \sqrt{X^{6}+4} \] So, the answer is: \[ \sqrt{X^{6} + 4} \]
