Dec 5, 2003 · 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, …

Oct 22, 2006 · The integral of cos (x²) from 0 to infinity is equal to the integral of sin (x²) over the same limits, both yielding the result of (1/2)√ (π/2). This conclusion is derived using complex …

Sep 30, 2021 · The integral of the function ∫ 1 x 2 + 2 d x is evaluated using trigonometric substitution. The correct approach involves recognizing that this integral can be transformed …

Sep 21, 2004 · Hi guys, I am looking in my calculus book, and it tells me in the example problem that the Integral of e^(-x)dx = -e^(-x) I don't see how u get this answer. I know the integral of …

Nov 9, 2006 · 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}^ …

Feb 23, 2012 · The integral of 1/z over the unit circle is equal to 2πi, contrary to the intuitive expectation that it should cancel out due to symmetry. This discrepancy arises because …

Jun 13, 2012 · Since you are integrating with respect to y, r^2 - x^2 = t^2 is a constant. So the integral becomes, \int_ {-t}^ {t} \sqrt {t^2-y^2}dy Now, the function you are integrating can be …

Jun 27, 2025 · Prove ∫ 0 2 / 4 1 x x 2 arcsin (x 1) (x 1 + x 9 16 x) 1 2 x d x = π 2 8 Let The representation integral of is Plugging identity above into with , we obtain Since the integrand is …

Jul 24, 2005 · One can use the Riemann-Stieltjes integral, limits, or Dirac sequences, or distribution theory to justify the Dirac Delta function. One can then deal with bothersome …

Dec 13, 2010 · This discussion focuses on solving definite integrals with variable limits, specifically addressing the derivative of integrals where both limits are functions of x. The …