Homework Statement
For the differential equation, verify (by differentiation and substitution) that the given function y(t) is a solution.Homework Equations
y' - 4ty = 1
y(t) = \int_{0}^{t} e^{-2(s^{2}-t^{2})} ds
The Attempt at a Solution
I attempted to take \frac{d}{dt} of y(t) as usual...
Help with Gaussian integration problem please :)
Homework Statement
Compute the improper integral
\int^{\infty}_{-\infty}x^{2}e^{-x^2}dx
given
\int^{\infty}_{-\infty}e^{-x^2}dx=\sqrt{\pi}.
Homework Equations
Just the rule for doubly-improper integrals I guess...
Homework Statement
Prove the convergence or divergence of the series \Sigma(\frac{n}{2n+3})^{2} using the Direct Comparison Test.
Homework Equations
If series A converges and every term in series B is less than the corresponding term in series A, then series B converges.
If series C...
Homework Statement
Find the integral of cot^6(x) without using the reduction formula.
Homework Equations
Potentially any trig identities involving the cotangent
The Attempt at a Solution
I tried splitting up the cot^6 in various ways such as cot^4*cot^2 and cot^2*cot^2*cot*2 but...
Homework Statement
Find the area of the lemniscate r2=6cos(2w) located to the right of the line r=3/2sec(w).
Homework Equations
Area of a polar region is the integral from a to b (in this case a and b are where the curve intersects the line, I believe) of 1/2*r2d(w).
The Attempt at...
Homework Statement
Convert the following Cartesian equation to polar form.
x^2/9 + y^2/4 = 1
Homework Equations
r*cos(t)=x
r*sin(t)=y
r=Sqrt(x^2 + y^2)
y/x = Arctan(t)
The Attempt at a Solution
I get ugly looking things like r^2(cos^2(t)/9 + sin^2(t)/4) = 1 but being a simple ellipse (edit...