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!)