Integral of [1-(e^(t/a))]^2 from a LR circuit

[harshasunder]

[SOLVED] integral problem

Homework Statement

hi. my problems an integral which i can't solve. the integral is this-

integral of [1-(e^(t/a))]^2 dt

a is a constant. it came when i tried to find the total power through a LR circuit

Homework Equations

The Attempt at a Solution

not sure at all.

[1-(e^(t/a))]^3
----------------
3 [ then? ]

 

[gamesguru]

This can be done with substitution:
$$\int(1-{e}^{t/a})^{2}dt$$
Let $$u=t/a$$ and thus $$du=dt/a, dt={a}{du}$$.
After expanding the integrand and putting $$a$$ (the constant) on the outside, it becomes:
$${a}\int{e}^{2u}-2e^u+1 du$$.
Again, if you don't see it already, use substitution on the first part and manually do the other two easy parts.
Let $$w=2u, dw=2du, \frac{1}{2}dw=du$$
$$\frac{1}{2}a\int{e^w}dw=\frac{1}{2}ae^w=\frac{1}{2}ae^{2u}=\frac{1}{2}ae^{2t/a}$$
The second part is easy:
$$-a \int{2e^u}du=-2e^u=-2e^{t/a}$$
The third part is the easiest (I added the constant 'C' here):
$$a \int du=au+C=a\frac{t}{a}+C=t+C$$ don't forget the substitution!
Summing all of these gives these final answer:
$$\int(1-{e}^{t/a})^{2}dt=\frac{1}{2}ae^{2t/a}-2e^{t/a}+t+C$$

 

[harshasunder]

hey thanks! a lot. how did you get the integral signs? (the big s) and the proper format?

 

[harshasunder]

hey thanks! a lot. how did you get the integral signs? (the big s) and the proper format?

 

[Schrodinger's Dog]

hey thanks! a lot. how did you get the integral signs? (the big s) and the proper format?

$\LaTeX$ is what you're looking for go to

https://www.physicsforums.com/showthread.php?t=8997

$$\int_0^{\infty} e^{x^2}\;dx\rightarrow \int \mid e^{x^2}\mid \lim_{x\rightarrow\infty}=$$

$$\sum_{n=-\infty}^\infty \frac{x^n}{n!} = \lim_{n\rightarrow\infty} (1+x/n)^n$$

You can click on any bit of code and cut and paste it as well. Try these ones.