安装免费的单位换算表单位转换工具程序!
安装免费的单位换算表单位转换工具程序!
安装免费的单位换算表单位转换工具程序!
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安装免费的单位换算表单位转换工具程序!
- What is the Integral of 2^ (x)? - Physics Forums
The integral of 2^x with respect to x is (1 ln (2)) * 2^x + C, where C is the constant of integration This result is derived using the property that the derivative of a^x is ln (a) * a^x, allowing for the straightforward application of integration techniques By substituting 2^x with e^ (x * ln (2)), the integration process simplifies significantly Understanding these principles is essential for solving similar exponential integrals Understanding of basic calculus concepts, particularly
- Solving Integrals for e^-ax^2: (i), (ii) (iii) - Physics Forums
The discussion focuses on solving integrals of the form \ (\int_ {0}^ {\infty} e^ {-ax^2} x^n dx\) for \ (n = 2, 3, 4\) using differentiation and integration by parts The integral \ (\int_ {0}^ {\infty} e^ {-ax^2} dx\) is established as \ (\frac {\sqrt {\pi}} {2\sqrt {a}}\) Participants suggest using differentiation with respect to the parameter \ (a\) for integrals (i) and (iii), while integration by parts is recommended for (ii) The final results for the integrals are \ (\int_ {0
- Finding Volume Under Cone Above Disk - Physics Forums
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 correct limits for integration in polar coordinates are established as r from 0 to 2 and θ from 0 to 2π The initial attempt incorrectly used r from -2 to 2, leading to an erroneous volume calculation of 32π 3 instead of the correct 16π 3 The need for understanding polar coordinates is emphasized for
- Integral of Sqrt(x)*e^-x: Step-by-Step Solution - Physics Forums
The integral ∫0∞ √ (x) * e^ (-x) dx evaluates to (√π) 2 The solution involves a substitution u = x^ (1 2), leading to the integral 2 ∫ e^ (-u^2) u^2 du By applying integration by parts and recognizing the integral of e^ (-u^2) as a known result, the final answer is confirmed as (√π) 2 The discussion clarifies the correct interpretation of limits and the application of the integration by parts formula Understanding of integration techniques, specifically integration by parts
- Solve Surface Integral: r^2 sin(theta) - Physics Forums
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 The integral is defined as integral E da = integral (1 4pi Eo) (q r^2) (r^2 sin (theta) d (theta) d (phi)) The participants clarify that this is a double integral with limits for both theta (0 to pi) and phi (0 to 2pi), which are determined by spherical coordinates The integration process for both variables is essential
- How can I solve integrals of the form x^n e^ (-x^2) by hand?
This discussion focuses on solving integrals of the form \int x^n e^ {-x^2} dx, particularly in the context of quantum mechanics The integral for n=2 simplifies to \int_ {-\infty}^ {\infty} x^2 e^ {-x^2} dx = \frac {\sqrt {\pi}} {2}, utilizing integration by parts and symmetry arguments The discussion highlights that odd powers of x^n yield zero due to the odd nature of the integrand, while higher even powers can be computed by reducing the power through integration by parts Additionally
- Integral Proof: u^2-a^2 | Detailed Explanation - Physics Forums
The integral proof discussed is for the expression \int du (u^2-a^2) = (1 2a) ln (u+a) (u-a) + C The solution involves trigonometric substitution, specifically using u=a*sec y, leading to the transformation of the integral into a solvable form Participants highlighted the importance of differentiating the right-hand side to verify the solution, emphasizing algebraic manipulation and properties of logarithms to achieve the final result The method of partial fraction decomposition was
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