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Integrate the function
$x \sqrt{1+2 x^2}$
$x \sqrt{1+2 x^2}$
Solution:
1942 Upvotes
Verified Answer
Let $1+2 x^2=t^2 \Rightarrow 4 x d x=2 t d t$
$\begin{aligned}
&\Rightarrow x d x=\frac{t}{2} d t \\
&\int x \sqrt{1+2 x^2} d x \\
&=\frac{1}{2} \int t^2 d t=\frac{t^3}{6}+\mathrm{C}=\frac{1}{6}\left(1+2 x^2\right)^{3 / 2}+\mathrm{C}
\end{aligned}$
$\begin{aligned}
&\Rightarrow x d x=\frac{t}{2} d t \\
&\int x \sqrt{1+2 x^2} d x \\
&=\frac{1}{2} \int t^2 d t=\frac{t^3}{6}+\mathrm{C}=\frac{1}{6}\left(1+2 x^2\right)^{3 / 2}+\mathrm{C}
\end{aligned}$
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