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    Integrals

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    Quick Calculation of Laplace Integral

    Please see the attached file for the fully formatted problem. Construct the quickest method to calculate the Laplace Integral. I = S e^(-x^2) dx infinity --> infinity

    Evaluating Integral Functions

    I'm taking a DE calculus class and I'm having problems figuring out the logic in solving some of the problems. The given integral is improper because both the interval of integration is unbounded and the integrand is unbounded near zero. Investigate its convergence by expressing it a sum of two intergrands-one from 0 to 1 an

    Double Integration Transformation

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

    Integrals: Cost function and Marginal Cost

    Given ist the following cost function: k(x)=x^3-9x^2+29x+35 x= quantity k= cost Question 1: At what quantity is the minimum of the marginal cost? Question 2: What is the increase of cost if the production is increased from 3 to 4 (integral)?

    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

    Fluid force - Applications of Integration

    NOTE: We are supposed to find the fluid force using integrals. I have attached a word document with the fluid force formula we are supposed to use. Please use the US system of measurement (i.e. pounds) Now here is the problem: A vertcial gate in a dam has the shape of an isoceles trapezoid. The top of the gate measures 10

    Integrate a natural log standing alone. How is that done

    The steps for integrating an ln standing alone are shown using the example Slnxdx. The same procedure can also be used for integrals of lns that can be simplified using the properties of logs such as ln3x, ln(x^2) or ln(square root of x), or if the entire ln is raised to a power.

    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.

    U substitution.

    The steps of U substitution are explained using the example of the integral of the square root of 4x+1.

    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.

    Integrate sin^2(x)?

    The steps for integrating sin^2(x) are shown with explanations of each step.

    U Substitution Example

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

    Integrals

    (a)Find the integral between 0 and infinity e^-st. sinat dt (b)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 in detail that f is not integrable on [0,1] but |f| is integrable

    Integrals Solutions Found

    (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)

    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: Boundaries and Infinity

    (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