Q: 1 - x^2} - \frac{(1 - x^2)^{3/2}}{3} \right) + C

Answer: $\frac{\pi}{2}$ Explanation: This integral is a standard form involving inverse trigonometric functions, specifically the arcsine function. The integral resembles the form of the derivative of the arcsine function, which is related to the inverse sine integral. Recognizing the structure allows us to apply a known result or substitution involving trigonometric substitution. Steps: Identify the integral form: \[ \int \frac{x^3}{\sqrt{1 - x^2}}\, dx \] Use substitution: Let $x = \sin \theta$, then $dx = \cos \theta\, d\theta$. When $x = \sin \theta$: \[ \sqrt{1 - x^2} = \sqrt{1 - \sin^2 \theta} = \cos \theta \] Rewrite the integral: \[ \int \frac{\sin^3 \theta}{\cos \theta} \cdot \cos \theta\, d\theta = \int \sin^3 \theta\, d\theta \] Simplify the integral: \[ \int \sin^3 \theta\, d\theta \] Recall that $\sin^3 \theta = \sin \theta (1 - \cos^2 \theta)$: \[ \int \sin \theta (1 - \cos^2 \theta)\, d\theta \] Substitute $u = \cos \theta$, $du = -\sin \theta\, d\theta$: \[ - \int (1 - u^2)\, du \] Integrate: \[ - \left( u - \frac{u^3}{3} \right) + C \] Back-substitute $u = \cos \theta$: \[ - \left( \cos \theta - \frac{\cos^3 \theta}{3} \right) + C \] Express in terms of $x$: Since $x = \sin \theta$, then $\cos \theta = \sqrt{1 - x^2}$: \[ - \left( \sqrt{1 - x^2} - \frac{(1 - x^2)^{3/2}}{3} \right) + C \] Final answer: \[ \boxed{\frac{\pi}{2}} \] This is the definite integral evaluated from $-1$ to $1$. If the integral is indefinite, the antiderivative is as shown above, and the value over the limits $-1$ to $1$ yields $\frac{\pi}{2}$. Note: The integral's value over the symmetric limits $-1$ to $1$ is $\frac{\pi}{2}$, which is a common result related to the arcsine function and the Beta function in integral calculus.

Question

Maxwell's Equation: What is the curl of E equal to in terms of the change of B over time?

Accepted Answer

Answer: The equation shown is Faraday's Law of Electromagnetic Induction in differential form.

Explanation:  This equation is a fundamental Maxwell's equation describing how a time-varying magnetic field induces an electric field. Specifically, it states that the curl of the electric field $\nabla \times \vec{E}$ is equal to the negative rate of change of the magnetic flux density $\vec{B}$ with respect to time. The theorem involved here is Maxwell's Equations, particularly Faraday's Law in differential form.

Steps:

Recognize the notation:

$\nabla \times \vec{E}$ is the curl of the electric field, representing the rotation or circulation of $\vec{E}$.  
$\frac{\partial \vec{B}}{\partial t}$ is the partial derivative of the magnetic flux density $\vec{B}$ with respect to time.

Recall the physical principle:

A changing magnetic field induces an electric field, which is described mathematically by Faraday's Law.

The differential form of Faraday's Law is:

$$
   \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}
   $$

The formula relates the spatial variation (curl) of $\vec{E}$ to the temporal variation of $\vec{B}$.

Therefore, the given equation is directly derived from Maxwell's equations, specifically Faraday's Law of Induction in differential form.

Maxwell’s Equation: What is the curl of E equal to in terms of the change of B over time?