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Integrals

Double Integration

Evaluate the double integral Transform the double integral of (i) using plane polar coordinates Show that the 3 x 3 determinant See attached file:

Evaluate integrals

Please see the attached file for the fully formatted problems. Evaluate the following integrals. S (4x^3 -2x - (2/x^3) dx S (1/2x^1/2) dx 1-->0 S ln x dx

Integration: Integration by Parts

I am trying to integrate e to a variable power times sin or cos using integration by parts, but I seem to be going in circles. How is this problem solved? The trick for solving e times sin or cos is shown using the example Se^x*sinxdx.

Integration of cos^2(x)

How do I integrate cos^2(x)? Please help me with this and include explanations so I can understand it.

U substitution

U substitution is explained using the example S4x(x^2+1)^5dx without and with limits of integration.

Integrals

(a) Find the integral between o and infinity (upper)of e^-x^2 dx . Use the above to prove that T(/2)= sqare root of pi where T represents the gamma function (b) Find the integral of x^3.e^-x^2 dx between the boundaries 0 and infinity (upper) thank you

Integration of a function

(a) let f:[0,1] ---R be the function f(x) = { x when x is an element of rational numbers {-x when x is not an element of rational numbers Prove that f is not integrable on [0,1] but |f| is integrable (b) Find the limit as x goes to 0 of 1/x the integral of e^t^2 dt between the boundaries 0 and x, x bei

Integrals

(a)Let f:[a,b] ---R be continuous and f(x)>= 0 for all x an element of [a,b]. prove that if the integral between the boundaries b and a of f(x) dx =0 then f(x) =0 for all x an element of [a,b] (b)Prove that the integral between infinity and 0 of e^-st .sinat dt = a/s^2 + a^2

Complex integrals

(1) let f:C----R be an analytic function such that f(1)=1. Find the value of f(3) (2) Evaluate the integral over & of dz/ z^2 -1 where & is the circle |z-i|=2 (3)Evaluate the integral over & of (z-1/z) dz where & is the line path from 1 to i (4) Evaluate the integral between 2pi and 0 of e^-i@ . e ^e^i@ d@