patterns in integrals

[someone]

hi everyone plz i need help with these three integrals :S

1st: how can i want the solution of this integral by partial fraction ?
integral 1/((x+a)(x+b)) dx

2nd: use integration by parts to solve this one and for what values of n is it valid?
integral x^n lnx dx

3rd : use tabular integration to solve
integral x^6 e^x dx

 

[galactus]

1st: how can i want the solution of this integral by partial fraction ?
$\displaystyle \int\frac{1}{(x+a)(x+b)}dx$

$\displaystyle \int\frac{1}{(x+a)(x+b)}dx=\int\frac{1}{(a-b)(x+b)}dx-\int\frac{1}{(a-b)(x+a)}dx$

The a-b is a constant, so you can move that outside the integral sign. Then you have an ln in your future.

2nd: use integration by parts to solve this one and for what values of n is it valid?
$\displaystyle \int{x^{n}ln(x)}dx$

Let $\displaystyle u=ln(x), \;\ dv=x^{n}dx, \;\ du=\frac{1}{x}dx, \;\ v=\frac{x^{n+1}}{n+1}$

3rd : use tabular integration to solve
$\displaystyle \int{x^{6}e^{x}}dx$

I am sorry to say, I do not know what tabular integration is. I reckon I could google it.

 

[galactus]

I remember what tabular integration is now.

Take successive derivatives of x^6 until it gets down to where you can not differentiate anymore. Add them up and attach the e^x(alternate the signs).

It's an easy way to do these types of problems.

$\displaystyle f(x)=x^{6}, \;\ f'(x)=6x^{5}, \;\ f''(x)=30x^{4}.........$ and so forth.

$\displaystyle (x^{6}-6x^{5}+30x^{4}-12-x^{3}+360x^{2}-720x+720)e^{x}$

See how that is done?. I had to recollect. Been a while since I used it. It's easy when you have something like x^6 to work with.

There is also a cool way called integration by recognition one can use.