1. ## integral

Evaluate the following integrals. z represent integral sign:-
(a)
Z xe¡x dx.
(b)
Z x(4 + x2)^10 dx. (Hint: use the substitution u = 4 + x2.)

2. Originally Posted by trythis
Evaluate the following integrals. z represent integral sign:-
(a)
Z xe¡x dx.
(b)
Z x(4 + x2)^10 dx. (Hint: use the substitution u = 4 + x2.)

I'm sorry, what does the i represent in your first integral? Did you mean $\displaystyle \int xe^{x}dx$ ????

If so use integration by parts!

$\displaystyle \int xe^x dx = [x \times \int e^x dx] - \int(( \int e^x dx) \times (\frac{d}{dx} x) dx)+C$

$\displaystyle = [xe^x ] - \int e^x dx )+C$

$\displaystyle = xe^x - e^x +C$

$\displaystyle = e^x(x-1)+C$

For the second one:

$\displaystyle u = 4+x^2$

Hence

$\displaystyle \frac{du}{dx} = 2x$

Hence $\displaystyle \frac{1}{2}du = xdx$

Now you can replace your original integral :

$\displaystyle \int x(4+x^2)^{10} dx$

$\displaystyle = \int (4+x^2)^{10} xdx$

$\displaystyle = \int (u)^{10} \frac{1}{2}du$

$\displaystyle = \frac{1}{2}\int (u)^{10} du$

You can do this, no? (remember to change back to original variable once you've integrated!)