MHB How to Integrate (x^2 + 2x + 1)sqrt(4x^2 + 8x + 5)?

[annie122]

how do i integrate (x^2 + 2x + 1)sqrt(4x^2 + 8x + 5)??

 

[mente oscura]

Re: integration

how do i integrate (x^2 + 2x + 1)sqrt(4x^2 + 8x + 5)??

Hello.

It is possible, that there is an easier way of solving the integral. But it could also be determined:

\displaystyle \int \ (x^2+2x+1) \sqrt{4x^2+8x+5}

\displaystyle \int \ (x+1)(x+1) \sqrt{4(x+1)^2+1}

u=x+1

v'=(x+1) \sqrt{4(x+1)^2+1}\displaystyle \ (uv)'=u'v+uv'

\displaystyle \int \ (uv)'= \int \ u'v+ \int \ uv'

Therefore:

\displaystyle \int \ uv' = uv- \int \ u'v

Try to follow.

Regards.

 

[annie122]

thanks, i can solve it now :)

 

[Prove It]

Re: integration

Hello.

It is possible, that there is an easier way of solving the integral. But it could also be determined:

\displaystyle \int \ (x^2+2x+1) \sqrt{4x^2+8x+5}

\displaystyle \int \ (x+1)(x+1) \sqrt{4(x+1)^2+1}

You have \displaystyle \begin{align*} \int{ \left( x + 1 \right) ^2 \sqrt{ 4 \left( x + 1 \right) ^2 + 1 } \, dx } \end{align*}, I would lean towards doing a substitution here...

\displaystyle \begin{align*} x + 1 = \frac{1}{2}\tan{\left( \theta \right) } \implies dx = \frac{1}{2} \sec^2{ \left( \theta \right) } \, d\theta \end{align*} and the integral becomes

\displaystyle \begin{align*} \int{ \left( x + 1 \right) ^2 \sqrt{4 \left( x + 1 \right) ^2 + 1 } \, dx} &= \int{ \left[ \frac{1}{2}\tan{ \left( \theta \right) } \right] ^2 \sqrt{ 4 \left[ \frac{1}{2} \tan{ \left( \theta \right) } \right] ^2 + 1 } \, \frac{1}{2} \sec^2{ \left( \theta \right) } \, d\theta } \\ &= \frac{1}{2} \int{ \frac{1}{4}\tan^2{ \left( \theta \right) } \sec^2{ \left( \theta \right) } \, \sqrt{ \tan^2{ \left( \theta \right) } + 1 } \, d\theta } \\ &= \frac{1}{8} \int{ \tan^2{\left( \theta \right) } \sec^2{ \left( \theta \right) } \sec{\left( \theta \right) } \,d\theta } \\ &= \frac{1}{8} \int{ \left[ \sec^2{\left( \theta \right) } - 1 \right] \sec^3{ \left( \theta \right) } \,d\theta } \\ &= \frac{1}{8} \int{ \sec^5{ \left( \theta \right) } - \sec^3{ \left( \theta \right) } \, d\theta } \\ &= \frac{1}{8}\int{ \frac{1}{\cos^5{ \left( \theta \right) } } - \frac{1}{\cos^3{ \left( \theta \right) } } \, d\theta } \\ &= \frac{1}{8} \int{ \frac{\cos{ \left( \theta \right) } }{\cos^6{ \left( \theta \right) } } - \frac{\cos{ \left( \theta \right) } }{\cos^4{ \left( \theta \right) } } \, d\theta } \\ &= \frac{1}{8} \int{ \cos{ \left( \theta \right) } \left\{ \frac{1}{\left[ 1 - \sin^2{ \left( \theta \right) } \right] ^3 } - \frac{ 1}{\left[ 1 - \sin^2{ \left( \theta \right) } \right] ^2 } \right\} \, d\theta } \end{align*}

Now let \displaystyle \begin{align*} u = \sin{ \left( \theta \right) } \implies du = \cos{ \left( \theta \right) } \, d\theta \end{align*} and the integral becomes

\displaystyle \begin{align*} \frac{1}{8} \int{ \cos{ \left( \theta \right) } \left\{ \frac{1}{\left[ 1 - \sin^2{ \left( \theta \right) } \right] ^3} - \frac{1}{ \left[ 1 - \sin^2{ \left( \theta \right) } \right] ^2 } \right\} \,d\theta } &= \frac{1}{8} \int{ \frac{1}{ \left( 1 - u^2 \right) ^3 } - \frac{1}{ \left( 1 - u^2 \right) ^2 } \, du } \\ &= \frac{1}{8} \int{ \frac{1}{ \left( 1 - u \right) ^3 \left( 1 + u \right) ^3 } - \frac{1}{ \left( 1 - u \right) ^2 \left( 1 + u \right) ^2 } \, du } \end{align*}

Which can be solved using partial fractions.Hmmm, that was a little too messy for my taste. Maybe a different original substitution would have been easier. Try \displaystyle \begin{align*} x + 1 = \frac{1}{2}\sinh{ (t) } \implies dx = \frac{1}{2}\cosh{(t)}\,dt \end{align*} and the integral becomes

\displaystyle \begin{align*} \int{ \left( x + 1 \right) ^2 \, \sqrt{4 \left( x + 1 \right) ^2 + 1 } \, dx } &= \int{ \left[ \frac{1}{2} \sinh{(t)} \right] ^2 \sqrt{ 4 \left[ \frac{1}{2}\sinh{(t)} \right] ^2 + 1 } \, \frac{1}{2}\cosh{(t)}\,dt } \\ &= \frac{1}{8} \int{ \sinh^2{(t)}\cosh{(t)} \, \sqrt{ \sinh^2{(t)} + 1 } \, dt } \\ &= \frac{1}{8} \int{ \sinh^2{(t)}\cosh^2{(t)} \, dt } \\ &= \frac{1}{8} \int{ \left[ \frac{1}{2} \sinh{(2t)} \right] ^2 \, dt } \\ &= \frac{1}{32} \int{ \sinh^2{(2t)} \, dt } \\ &= \frac{1}{32} \int{ \frac{1}{2} \left[ \cosh{(4t)} - 1 \right] \, dt } \\ &= \frac{1}{64} \int{ \cosh{(4t)} - 1 \, dt } \\ &= \frac{1}{64} \left[ \frac{1}{4}\sinh{(4t)} - t \right] + C \\ &= \frac{1}{256}\sinh{(4t)} - \frac{1}{64}t + C \\ &= \frac{1}{128}\sinh{(2t)}\cosh{(2t)} - \frac{1}{64}t + C \\ &= \frac{1}{64}\sinh{(t)}\cosh{(t)} \left[ 2 \sinh^2{(t)} + 1 \right] - \frac{1}{64}t + C \\ &= \frac{1}{64} \sinh{(t)} \left[ 2 \sinh^2{(t)} + 1 \right] \, \sqrt{ \sinh^2{(t)} + 1 } - \frac{1}{64}t + C \\ &= \frac{1}{32} \left( x + 1 \right) \left\{ 2 \left[ 2 \left( x + 1 \right) \right] ^2 + 1 \right\} \, \sqrt{ \left[ 2 \left( x + 1 \right) \right] ^2 + 1 } - \frac{1}{64}\,\textrm{arsinh}\, { \left[ 2 \left( x + 1 \right) \right] } + C \\ &= \frac{1}{32} \left( x + 1 \right) \left[ 8 \left( x + 1 \right) ^2 + 1 \right] \, \sqrt{ 4 \left( x + 1 \right) ^2 + 1 } - \frac{1}{64} \, \textrm{ arsinh } \, { \left[ 2 \left( x + 1 \right) \right] } + C \end{align*}

PHEW!

 

[soroban]

Hello, Yuuki!

\int (x^2 + 2x + 1)\sqrt{4x^2 + 8x + 5}\,dx

Under the radical, we have:
. . 4(x^2+2x+1) + 1 \:=\:4(x+1)^2+1

The integral becomes: .\int (x+1)^2\sqrt{4(x+1)^2+1}\,dx

Let u \,=\,x+1\quad\Rightarrow\quad dx \,=\,du

Then we have: .\int u^2\sqrt{4u^2+1}\,du

Let u \,=\,\tfrac{1}{2}\tan\theta \quad \Rightarrow\quad du \,=\,\tfrac{1}{2}\sec^2\!\theta\,d\theta

And we have: .\int\left(\tfrac{1}{2}\tan\theta\right)^2\left( \sec\theta\right)\left( \tfrac{1}{2}\sec^2\!\theta\,d\theta\right)

. . =\;\tfrac{1}{8}\int\sec^3\!\theta\tan^2\!\theta\,d\theta \;=\;\tfrac{1}{8}\int\sec^3\!\theta(\sec^2\!\theta -1)\,d\theta

. . =\;\tfrac{1}{8}\int(\sec^5\!\theta - \sec^3\!\theta)\,d\thetaApply the reduction formula:
. . \int \sec^n\!x\,dx \;=\;\frac{1}{2}\left[\sec^{n-2}\!x\tan x + \int\sec^{n-2}\!x\,dx\right]

and remember to back-substitute.