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The integral \ (\int_ {0}^. The discussion focuses on solving the surface integral of the expression r^2 sin (theta) d (theta) d (phi) in the context of magnetic flux e through a sphere of radius r. This discussion focuses on solving definite integrals with variable limits, specifically addressing the derivative of integrals where both limits are functions of x.

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Summary the integral is proven to equal through the application of fubini's theorem and trigonometric substitution. This result is derived using the property that the derivative of a^x is ln (a) * a^x,. The transformation simplifies the integral, allowing for.

The integral of 2^x with respect to x is (1/ln (2)) * 2^x + c, where c is the constant of integration.

Then write the integral of y 2 exp (. The discussion centers on calculating the volume of a solid under the cone defined by z = √ (x² + y²) and above the disk described by x² + y² ≤ 4 using polar coordinates. The solution involves trigonometric substitution, specifically using u=a*sec y,. I know the integral of.

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