Question

Prove that the integral from 0 to Infinity (sin^2(x)e^-x)dx converges using the comparison test.

Answer #1

definite integral 0 to 7 of e^(x)sin(x)dx

Determine whether the improper integral from 5 to infinity
3/square root x dx converges or diverges, and find the value if it
converges.
Select the correct choice below and fill in any answer boxes
within your choice.
A. The value of the integral
B.The integral diverges.

Explain whether the following integrals converge or not. If the
integral converges, find the value. If the integral does not
converge, describe why (does it go to +infinity, -infinity,
oscillate, ?)
i) Integral from x=1 to x=infinity of x^-1.4 dx
ii) Integral from x=1 to x=infinity of 1/x^2 * (sin x)^2 dx
iii) Integral from x=0 to x=1 of 1/(1-x) dx

Determine the convergence or divergence if each integral by
using a comparison function. Show work using the steps below:
A. Indicate the comparison function you are using.
B. Indicate if your comparison function is larger or smaller
than the original function.
C. Indicate if your comparison integral converges or diverges.
Explain why.
D. State if the original integral converges or diverges. If it
converges, you don’t need to give the value it converges to.
11. integral from 1 to infinity...

Integrate -infinity to -3 x/(x^2+x-2)dx if it
converges

using reduction formula***
13. Evaluate, ∫ sin 5x cos x dx. Also prove that, ∫ sin mx cos
nx dx = 0

a). Find dy/dx for the following integral.
y=Integral from 0 to cosine(x) dt/√1+ t^2 ,
0<x<pi
b). Find dy/dx for tthe following integral
y=Integral from 0 to sine^-1 (x) cosine t dt

Question about using the convolution of distribution:
1. we have the formula: integral fx(x)fy(z-x)dx=integral
fx(z-x)fy(x)dx
I know this are equivalent. However, how do I decide which side
I should use ?
For example,X~Exp(1) and Y~Unif [0,1] X and Y independnt and the
textbook use fx(z-x)fy(x)dx.
However, can I use the left hand side fx(x)fy(z-x)dx???is there
any constraint for using left or right or actually both can lead me
to the right answer???
2. For X and Y are independent and...

Question B:Consider the integral of sin(x) * cos(x) dx.
i) Do it using integration by parts; you might need the “break
out of the loop” trick. I would do u=sin(x), dv=cos(x)dx
ii) Do it using u-substitution. I would do u=cos(x)
iii) Do it using the identity sin(x)*cos(x)=0.5*sin(2x)
iv) Explain how your results in parts i,ii,iii relate to each
other.

Prove that for positive integers,
Integral from nothing to nothing tangent Superscript n Baseline
x dx equals StartFraction tangent Superscript n minus 1 Baseline x
Over n minus 1 EndFraction minus Integral from nothing to nothing
tangent Superscript n minus 2 Baseline x dx comma n not equals
1∫tannx dx=tann−1xn−1−∫tann−2x dx, n≠1.
Use the formula to evaluate
Integral from 0 to StartFraction pi Over 3 EndFraction 4 tangent
Superscript 5 Baseline x dx∫0π34tan5x dx.

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