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# Integrals

### Sketch the curve and set up double integral for bounded area.

Please see the attached file for the fully formatted problems. Sketch the curve r = 5 - 3cos(theta) and set up double integral for bounded area in the third quadrant.

### This problem involves evaluating an indefinite integral using the substitution rule.

Evaluate the following indefinite integral. int[(x^a)sqrt(r+tx^(a+1))]dx, (t not=0, a not=-1). (See attachment)

### Triple Integrals : Finding Volume of Solids with Boundaries

1) Evaluate the triple integral e^(1-(x^2)-(y^2)) dxdydz with T the solid enclosed by z=0 and z= 4-(x^2)-(y^2) 2) Find the volume of the solid bounded above and below by the cone (z^2) = (x^2) + (y^2), and the side by y=0 and y= square root(4-(x^2)-(z^2))

### 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

### Integration: Standard Partition, Integrable over a Range

Questions on integration, see attachment.

### Integral test: Convergence or divergence of a series.

Use the integral test to determine the convergence or divergence of the series: En=1 2 / (3n + 5)

### Integration

Integral x-1, divided by x to the 3rd + x squared to dx. x-1 ----- X to the 3rd + X to the 2nd ( all dx)

### Integration

For problem #1, its the integral from o to infinity (the symbol for infinity for that problem was cut off)

### Evaluating integrals

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

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

### Calculating integrals: Sines and Cosines

Integrate {cos(x)cos(3x)cos(5x),x}

### How do I integrate sin or cos to an even power greater than 2?

The steps for integrating sin or cos to an even power greater than 2 are shown using the example Ssin^4(x)dx.

### I am trying to 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.

### Question about please explain the process of U substitution

The steps of U substitution are explained using the example S(3x)/(5x^2-2).

### 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.

### What am I doing when I integrate a function? (with no numbers at the top and bottom of the integration symbol)

Integration is explained using the example Sx^2dx.

### Working with initial value problems.

Solve the initial value problem: d²y/dx²= sec²x y(0)=0 and y'(0)=1

### To solve the problem by separation of variables

I have no idea how to start this problem. dN/dt + N = Nte^t+2 So far I have, dN/dt = Nte^t+2 - N where do I go from there.

### 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@

### Euler's and improved Euler's method,...find the exact solution

1. y'=1/2 -x + 2y y(0) = 1 Find the exact solution "Sphee" {0 with line diagonal of the 0} 2. y'=x square + y square y(0)=1 a. Let h=.1 use Euler and improve to approximate to get Sphee(.1

### Evaluate an indefinite integral.

Evaluate the indefinite integral: (x^2+2x-3)/(x^4)