Help with integration

**ryanali** · September 6th 2010, 11:42 AM

1) Integral {(x^3)-4x}/{(x^2)+1}^2.what Is The Solution?
2) Integral {x^(1/2)}dx/{1+ X^(1/4)}.what Is The Solution?

Please help me out.i require detailed solutions.
thaking you.

**mr fantastic** · September 6th 2010, 01:01 PM

Originally Posted by ryanali

1) Integral {(x^3)-4x}/{(x^2)+1}^2.what Is The Solution?
2) Integral {x^(1/2)}dx/{1+ X^(1/4)}.what Is The Solution?

Please help me out.i require detailed solutions.
thaking you.

1) Use partial fractions.

2) Substitute u = x^1/4.

If you need more help, please show your effort and say where you're stuck.

**TheCoffeeMachine** · September 6th 2010, 01:04 PM

1. $\displaystyle \int \frac{x^3-4x}{(x^2+1)^2}\;{dx}$ $\displaystyle = \int \frac{x(x^2-4)}{(x^2+1)^2}\;{dx}$

Let $\displaystyle t = x^2+1 \Rightarrow dx = \frac{1}{2x}\;{dt}$

Then also $x^2-4 = t-5$

We have $\displaystyle \int \frac{x^3-4x}{(x^2+1)^2}\;{dx}$ $\displaystyle = \frac{1}{2}\int \frac{x(t-5)}{xt^2}\;{dt} = \frac{1}{2}\int \left(\frac{1}{t}-\frac{5}{t^2}\right)\;{dt} = \ln{t}+\frac{5}{t}+k$

Hence $\displaystyle \int \frac{x^3-4x}{(x^2+1)^2}\;{dx} = \frac{(x^2+1)\ln(x^2+1)+5}{2(x^2+1)}+k$

**ryanali** · September 7th 2010, 04:22 AM

Thank you very much to you both for replying.I tried the first using partial fractions and substitution,but got stuck and the second one i still have no idea how to go forward.
Thank you TheCoffeeMachine.If it is is'nt too much for you could you please help me with the second one too.

**tom@ballooncalculus** · September 7th 2010, 07:25 AM

Just in case a picture helps...

In fact, a choice of two - either...

... or...

... where (key in spoiler) ...

Spoiler:

Either way, work anti-clockwise, and find G then F then I.

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

**mr fantastic** · September 7th 2010, 12:40 PM

Originally Posted by ryanali

Thank you very much to you both for replying.I tried the first using partial fractions and substitution,but got stuck and the second one i still have no idea how to go forward.
Thank you TheCoffeeMachine.If it is is'nt too much for you could you please help me with the second one too.

You should post all the work you have done and then say specifically where you are stuck.

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