Author Topic: Integration Q's (Read 747 times) Tweet Share

Truck · « **on:** June 25, 2012, 11:29:45 pm »

Hey, really struggling with this specific integral:

$\int \sqrt{1+\sqrt{x}}, dx$

Would greatly appreciate help on how to do it, thanks.

EDIT: Finally fixed the LaTeX, took me a while lol.

b^3 · « **Reply #1 on:** June 25, 2012, 11:43:28 pm »

$\begin{alignedat}{1}\int\sqrt{1+\sqrt{x}} & dx\end{alignedat}$
Let $x=\sqrt{u}$
$\begin{alignedat}{1}\frac{du}{dx} & =\frac{1}{2}x^{-\frac{1}{2}}\end{alignedat}=\frac{1}{2u}$
$\begin{alignedat}{1}\int\sqrt{1+\sqrt{x}}dx & =\int\sqrt{1+u}\times2udu \\ & =2\int u\sqrt{1+u}du \end{alignedat}$
Let $a=1+u$
$\begin{alignedat}{1}\frac{da}{du} & =1\end{alignedat}$ and $u=a-1$
$\begin{alignedat}{1}2\int u\sqrt{1+u}du & =2\int\left(a-1\right)a^{\frac{1}{2}}da \\ & =2\int a^{\frac{3}{2}}-a^{\frac{1}{2}}da \\ & =2\left(\frac{2}{5}a^{\frac{5}{2}}-\frac{2}{3}a^{\frac{3}{2}}\right)+C \\ & =\frac{4}{5}(u+1)^{\frac{5}{2}}-\frac{4}{3}\left(u+1\right)^{\frac{3}{2}}+C \\ & =\frac{4}{5}(\sqrt{x}+1)^{\frac{5}{2}}-\frac{4}{3}\left(\sqrt{x}+1\right)^{\frac{3}{2}}+C \\ & =\frac{4}{15}\left(\sqrt{x}+1\right)^{\frac{3}{2}}\left(3\left(\sqrt{x}+1\right)-5\right)+C \\ & =\frac{4}{15}\left(\sqrt{x}+1\right)^{\frac{3}{2}}\left(3\sqrt{x}-2\right)+C \end{alignedat}$

Hope that helps

Truck · « **Reply #2 on:** June 25, 2012, 11:52:59 pm »

Quote from: b^3 on June 25, 2012, 11:43:28 pm

$\begin{alignedat}{1}\int\sqrt{1+\sqrt{x}} & dx\end{alignedat}$
Let $x=\sqrt{u}$
$\begin{alignedat}{1}\frac{du}{dx} & =\frac{1}{2}x^{-\frac{1}{2}}\end{alignedat}=\frac{1}{2u}$
$\begin{alignedat}{1}\int\sqrt{1+\sqrt{x}}dx & =\int\sqrt{1+u}\times2udu \\ & =2\int u\sqrt{1+u}du \end{alignedat}$
Let $a=1+u$
$\begin{alignedat}{1}\frac{da}{du} & =1\end{alignedat}$ and $u=a-1$
$\begin{alignedat}{1}2\int u\sqrt{1+u}du & =2\int\left(a-1\right)a^{\frac{1}{2}}da \\ & =2\int a^{\frac{3}{2}}-a^{\frac{1}{2}}da \\ & =2\left(\frac{2}{5}a^{\frac{5}{2}}-\frac{2}{3}a^{\frac{3}{2}}\right)+C \\ & =\frac{4}{5}(u+1)^{\frac{5}{2}}-\frac{4}{3}\left(u+1\right)^{\frac{3}{2}}+C \\ & =\frac{4}{5}(\sqrt{x}+1)^{\frac{5}{2}}-\frac{4}{3}\left(\sqrt{x}+1\right)^{\frac{3}{2}}+C \\ & =\frac{4}{15}\left(\sqrt{x}+1\right)^{\frac{3}{2}}\left(3\left(\sqrt{x}+1\right)-5\right)+C \\ & =\frac{4}{15}\left(\sqrt{x}+1\right)^{\frac{3}{2}}\left(3\sqrt{x}-2\right)+C \end{alignedat}$

Hope that helps

Definitely does! Thanks bcub3d. I was stuck on the 4th line, didn't realize you could make dx = 2u du and thus just substitute that into the equation!

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Author Topic: Integration Q's (Read 747 times) Tweet Share

Truck

Integration Q's

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