Reduction formula question (int by parts)

[thomas49th]

Homework Statement

Let $$I_{n} = \int^{2}_{0} x^{n}e^{x} dx$$ where n is a positive integer. Use integration by parts to show
that

$$2^{n}e^{2} - nI_{n-1}$$

By first finding

$$I_{1} = \int ^{2}_{0} xe^{x} dx$$
find I2 and I3.

Homework Equations

I'm sure your all aware of the formula for Int by parts. We'll take the $$e^{x}$$ function as the one to integrate and the $$x^{n}$$ as the one to differentiate.

The Attempt at a Solution

So use integration by parts to find In:

$$x^{n}e^{x} - n\int^{2}_{0}x^{n-1}$$

Well that's
$$x^{n}e^{x} - nI_{n-1}$$
is it not?

So now put the limits in

$$[x^{n}e^{x} - nI_{n-1}]^{2}_{0}$$

$$[2^{n}e^{2} - nI_{n-1}] - [ - nI_{n-1}]$$
but that's isn't right because I'm getting no $$nI_{n-1}$$ because they cancel!

Where have I gone wrong?

Thanks
Thomas

 

[Gib Z]

So use integration by parts to find In:

$$x^{n}e^{x} - n\int^{2}_{0}x^{n-1}$$

Well that's
$$x^{n}e^{x} - nI_{n-1}$$
is it not?

The step is correct but neither of them represent $I_n$. The Integration by parts formula for definite integrals is $$\int^b_a u(x) \frac{dv(x)}{dx} dx = u(b)v(b)-u(a)v(a) - \int^b_a v(x) \frac{du(x)}{dx}$$.

You have put in the limits of integration for the second term, but not the first.

 

[Mark44]

Homework Statement

Let $$I_{n} = \int^{2}_{0} x^{n}e^{x} dx$$ where n is a positive integer. Use integration by parts to show
that

$$2^{n}e^{2} - nI_{n-1}$$

You omitted part of the statement above. Show that 2ⁿe² - nI_{n - 1} equals what or does what?

By first finding

$$I_{1} = \int ^{2}_{0} xe^{x} dx$$
find I2 and I3.

Homework Equations

I'm sure your all aware of the formula for Int by parts. We'll take the $$e^{x}$$ function as the one to integrate and the $$x^{n}$$ as the one to differentiate.

The Attempt at a Solution

So use integration by parts to find In:

$$x^{n}e^{x} - n\int^{2}_{0}x^{n-1}$$

Well that's
$$x^{n}e^{x} - nI_{n-1}$$
is it not?

So now put the limits in

$$[x^{n}e^{x} - nI_{n-1}]^{2}_{0}$$

$$[2^{n}e^{2} - nI_{n-1}] - [ - nI_{n-1}]$$
but that's isn't right because I'm getting no $$nI_{n-1}$$ because they cancel!

Where have I gone wrong?

Thanks
Thomas