Author Topic: Integration...NEED HELP!!! PLZ!! (Read 979 times) Tweet Share

squance · « **on:** July 30, 2008, 07:39:50 pm »

These are some integration questions I recently faced in my first tutorial class at uni.
It was supposed to be revision and we were expected to know this...
except...I forgotten how to do these.

Can someone help please?
Solutions are at the bottom.
Thanks

(h) $\int\frac{1}{x^2+2x+6}dx$

(i) $\int\frac{x}{\sqrt{1+x}}dx$

(j) $\int\frac{x^4-3x^2}{x^5-5x^3-24}dx$

(k) $\int\frac{4x}{(2x-1)^2}dx$

[IMG]http://img167.imageshack.us/img167/794/img0001oq2.th.jpg[/img]

ell · « **Reply #1 on:** July 30, 2008, 08:00:57 pm »

Couple of hints:

(h) Complete the square for the denominator, $x^2 + 2x + 6 = x^2 + 2x + 1 + 5 = (x+1)^2 + 5$ . Then apply $\int \frac{a}{a^2 + x^2}\ dx = \tan^{-1} \frac{x}{a} + C$

(i) Use the substitution $u = 1 + x$

Then $x = u - 1$ and $\frac{du}{dx} = 1$

(j) Let $u = x^5 - 5x^3 - 24$

Then $\frac{du}{dx} = 5x^4 - 15x^2 = 5(x^4 - 3x^2)$

The new integral is now $\frac{1}{5} \int \frac{1}{u}\ du$

(k) Another linear substitution - hopefully you'll be able to get this one, it is very similar to (i).

squance · « **Reply #2 on:** July 30, 2008, 08:14:31 pm »

Thanks.
I've have a go of them now.

squance · « **Reply #3 on:** July 31, 2008, 11:51:56 am »

So for the first one...

(h) $\int\frac{1}{x^2+2x+6}dx$

completing square of denominator is $(x+1)^2+5$

so.... $\int\frac{1}{(x+1)^2+5}dx$

Let u = x + 1 du/dx = 1
x=u-1

so $\int\frac{1}{u^2+5}du$

= $\frac{1}{\sqrt5}arctan\frac{u}{\sqrt{5}}+c$

= $\frac{1}{\sqrt5}arctan\frac{x+1}{\sqrt{5}}+c$

(i) $\int\frac{x}{\sqrt{1+x}}dx$
Let u = 1 + x du/dx = 1
x = u-1

$\int\frac{u-1}{\sqrt{1+u-1}}du$
$\int\frac{u-1}{\sqrt{u}}du$
$\int u^\frac{1}{2} - u^\frac{-1}{2} du$
$\frac{2}{3} (u)^\frac{3}{2}-2(u)^\frac{1}{2}$
$\frac{2}{3} (1+x)^\frac{3}{2}-2(1+x)^\frac{1}{2}+c$

Is this correct?
(I can't be stuffed putting up the other two but I'll be happy if someone can tell me if these two are right...)

dcc · « **Reply #4 on:** July 31, 2008, 04:21:48 pm »

Both are correct

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Author Topic: Integration...NEED HELP!!! PLZ!! (Read 979 times) Tweet Share

squance

Integration...NEED HELP!!! PLZ!!

ell

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squance

Re: Integration...NEED HELP!!! PLZ!!

squance

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dcc

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