# Thread: Need help with 2 Integrals

1. ## Need help with 2 Integrals

Could u guys give me some tips about how to aproach these integrals please? http://i284.photobucket.com/albums/l...hreat/Mate.png (Img in attachment too.) Thanks Alot!

2. ## Re: Need help with 2 Integrals

Originally Posted by purpleclover
Could u guys give me some tips about how to aproach these integrals please? http://i284.photobucket.com/albums/l...hreat/Mate.png (Img in attachment too.) Thanks Alot!
\displaystyle \begin{align*} \frac{(x + 1)^3}{(x + 2)(x + 3)(x + 4)^2} = \frac{x^3 + 3x^2 + 3x + 1}{(x + 2)(x + 3)(x + 4)^2} \end{align*}

Now apply partial fractions with \displaystyle \begin{align*} \frac{A}{x + 2} + \frac{B}{x + 3} + \frac{C}{(x + 4)} + \frac{D}{(x + 4)^2} = \frac{x^3 + 3x^2 + 3x + 1}{(x + 2)(x +3)(x + 4)^2} \end{align*}

3. ## Re: Need help with 2 Integrals

Thanks alot!... How about the second one ?

4. ## Re: Need help with 2 Integrals

Originally Posted by purpleclover
Could u guys give me some tips about how to aproach these integrals please? http://i284.photobucket.com/albums/l...hreat/Mate.png (Img in attachment too.) Thanks Alot!
\displaystyle \begin{align*} \int{\frac{x+1}{\sqrt{x^2 + x + 1}}\,dx} &= \frac{1}{2}\int{\frac{2x + 2}{\sqrt{x^2 + x + 1}}\,dx} \\ &= \frac{1}{2}\int{\frac{2x + 1 + 1}{\sqrt{x^2 + x + 1}}\,dx} \\ &= \frac{1}{2}\int{\frac{2x + 1}{\sqrt{x^2 + x + 1}} + \frac{1}{\sqrt{x^2 + x + 1}}\,dx} \\ &= \frac{1}{2}\int{ \frac{2x+1}{\sqrt{x^2 + x + 1}} + \frac{1}{ \sqrt{ x^2 + x + \left( \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2 + 1} }\,dx} \\ &= \frac{1}{2}\int{\frac{2x+1}{\sqrt{x^2+x+1}} + \frac{1}{\sqrt{\left(x+\frac{1}{2}\right)^2 + \frac{3}{4}}}\,dx} \end{align*}

The first term can be integrated by making the substitution \displaystyle \begin{align*} u = x^2 + x + 1 \end{align*} and the second can be integrated by making the substitution \displaystyle \begin{align*} x + \frac{1}{2} = \frac{\sqrt{3}}{2}\sinh{t} \end{align*}.

5. ## Re: Need help with 2 Integrals

holy C***.... THANKS ALOT!

6. ## Re: Need help with 2 Integrals

i was woundering in the second integral instead of making the substituion x+1/2=sinht... cant i just make the substituion x+1=t...and then use the ln|..| formula?

7. ## Re: Need help with 2 Integrals

Originally Posted by purpleclover
i was woundering in the second integral instead of making the substituion x+1/2=sinht... cant i just make the substituion x+1=t...and then use the ln|..| formula?
No you can't. The ln|...| formula only works when the top is the derivative of the bottom, which it isn't.

k thanks.