What is the Connection Between Polar Coordinates and Integrals?

[Phymath]

two questions...

1) by a change of variable show the following...
$$\int^{\infty} _{-\infty} \frac{dt}{(a^2 + t^2)^{3/2}} = \frac{2}{a^2}\int^{\pi/2} _0 cos \ t \ dt$$

i'm thinking about changing this to polar cords and see where that take me anyone?

2) $$F(x) = \int^{cos \ x} _0 e^{xt^2} \ dt , \ G(x) = \int^{cos \ x} _0 t^2 e^{xt^2} \ dt, \ H(t) = G(x) - F'(x)$$ express H(x) in elementry functions.

$$F'(x) = \frac{\partial cos \ x}{\partial x} e^{xt^2} - 0 + \int \frac{\partial}{\partial x} e^{xt^2} \ dt \\<br /> = -sin \ x \ e^{xt^2} + \int t^2e^{xt^2} \ dt$$

$$H(x) = \int^{cos \ x} _0 t^2 e^{xt^2} \ dt + sin \ x \ e^{xt^2} - \int t^2e^{xt^2} \ dt <br /> <br /> = sin \ x \ e^{xt^2}$$

Did i do the dirv correctly? and how do I show that $$H(\pi/4) = \frac{e^{\pi/8}}{\sqrt{2}}$$

 

[Curious3141]

two questions...

1) by a change of variable show the following...
$$\int^{\infty} _{-\infty} \frac{dt}{(a^2 + t^2)^{3/2}} = \frac{2}{a^2}\int^{\pi/2} _0 cos \ t \ dt$$

i'm thinking about changing this to polar cords and see where that take me anyone?

Hint : try $$\frac{t}{a} = \tan \theta$$

 

[dextercioby]

For the second,u should plug $t\rightarrow \cos x$ and then it will be simple to find the final formula...

Daniel.

P.S.What polar coordinates...?It's not a double integral...

 

[Phymath]

why would i plug t in for cos x? do u mean t for x so its cos t? I am confused why i would do that

 

[Phymath]

i have to take the function at the upper limit don't I? yep yea i do, ok thanks all