Recent content by utkarshakash

The polar equation of given parabola will be ##r=-2a/(1+cos \theta)##. The limit of theta is pi/4 to pi/2. Integrating the expression wrt r I'm left with ##\int_{\pi/4}^{\pi/2} \dfrac{-2a cos 2 \theta}{1+cos \theta} d \theta ##

Because I'm required to solve it by changing into polar coordinates.

Homework Statement Evaluate the integral by changing into polar coordinates. \displaystyle \int_0^{4a} \int_{y^2/4a}^y \dfrac{x^2-y^2}{x^2+y^2} dx dy The Attempt at a Solution Substituting x=rcos theta and y=rsin theta , the integrand changes to cos 2 \theta r dr d \theta . I know that the...

@Pranav-Arora See post #8 of mine.

I appreciate your effort and patience for posting the complete solution. However, your method was too complex for me. I don't know anything about contours and related terms. The integral which I posted here is a part of another integral which forms the original problem and here's it. Calculate...

Yes

Homework Statement $$ \displaystyle \int_0^{\infty} e^{-x} \dfrac{a\sin ax - \cos ax}{1+a^2} da $$ Homework Equations The Attempt at a Solution Evaluating this using integration by parts will be a cumbersome process and I don't even think that would give me the answer. Substitutions aren't...

Homework Statement Evaluate \displaystyle \int_0^{\pi} \log (1+a\cos x) dx Homework Equations The Attempt at a Solution Using Leibnitz's Rule, F'(a)=\displaystyle \int_0^{\pi} \dfrac{\cos x}{1+a \cos x} dx Now, If I assume sinx=t, then the above integral changes to \displaystyle \int_0^{0}...

Thanks!

Got it! Thanks a lot!

I've already checked it thrice but couldn't find any error! Can you please show me where I'm going wrong?

Homework Statement Find the value of n so that the equation V=r^n(3 \cos ^3 \theta -1) satisfies the relation $$\dfrac{\partial}{\partial r} \left( r^2 \dfrac{\partial V}{\partial r} \right) + \dfrac{1}{\sin \theta}\dfrac{\partial}{\partial \theta} \left( \sin \theta \dfrac{\partial...

$$f(0) = f(x) -xf'(x)+\frac{x^2}{2!} f''(x) - \frac{x^3}{3!}f'''(x)+...$$ I still can't figure out what to do next.

Homework Statement IF e^{m \arctan x}=a_0 + a_1x + a_2x^2 + a_3x^3...+a_nx^n+... prove that (n+1)a_{n+1} + (n-1)a_{n-1}=ma_n and hence obtain the expansion of e^{m \arctan x} . Homework Equations The Attempt at a Solution $$e^{m \arctan x} = 1+m \arctan x + (m \arctan x)^2/2! + (m...