Question

Find $ \int{\sqrt{10-4x+4{{x}^{2}}}\text{ }dx} $

Answer 1

Hint: Integration of a given expression helps in returning to the expression which was differentiated to get integrated expression. To integrate quadratic expression, first the expression should be modified to get squared term and the constant term.

Complete step-by-step answer:
Expression whose integral has to be determined is given as:
 $ \int{\sqrt{10-4x+4{{x}^{2}}}\text{ }dx} $
The expression under square root is quadratic and to simplify it, the terms should be modified in such a way that square expression of x is obtained separated from a constant term. This is calculated as:
\[\begin{align}
  & \sqrt{10-4x+4{{x}^{2}}}=\sqrt{{{(2x)}^{2}}-2\times 2x\times 1+{{(1)}^{2}}+9} \\
 & =\sqrt{{{(2x-1)}^{2}}+{{(3)}^{2}}} \\
 & =\sqrt{{{(3)}^{2}}+{{(2x-1)}^{2}}}
\end{align}\]
In above expression, obtained the expression has been simplified to obtained desired form and now on this form, the formula of integration that when applied gives the answer is:
\[\int{\sqrt{{{a}^{2}}+{{x}^{2}}}}dx=\dfrac{1}{2}x\sqrt{{{a}^{2}}+{{x}^{2}}}+\dfrac{1}{2}{{a}^{2}}\log \left| x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right|+C\]
The expression, $ \int{\sqrt{10-4x+4{{x}^{2}}}\text{ }dx} $ when compared to \[\int{\sqrt{{{a}^{2}}+{{x}^{2}}}}dx\] gives “a” equal to $ 3 $ while x when compared becomes equal to $ (2x-1) $ .
The integration of expression, \[\sqrt{10-4x+4{{x}^{2}}}\]is calculated as:
  $ \begin{align}
  & \int{\sqrt{10-4x+4{{x}^{2}}}\text{ }dx}=\dfrac{1}{2}(2x-1)\sqrt{{{3}^{2}}+{{(2x-1)}^{2}}}+\dfrac{1}{2}{{(3)}^{2}}\log \left| (2x-1)+\sqrt{{{3}^{2}}+{{(2x-1)}^{2}}} \right|+C \\
 & =x\sqrt{10-4x+4{{x}^{2}}}-\dfrac{1}{2}\sqrt{10-4x+4{{x}^{2}}}+\dfrac{9}{2}\log \left| (2x-1)+\sqrt{10-4x+4{{x}^{2}}} \right|+C
\end{align} $
The above expression obtained is the primitive or anti-derivative of expression, $ \int{\sqrt{10-4x+4{{x}^{2}}}\text{ }dx} $ . The term C obtained is written to denote any constant number which was present in primitive function but due to derivative, it became equal to zero.
The derivative of primitive function obtained again gives the expression as f(x) dx which indicates that integration and derivative are related to each other.

Note: Another name of integration is primitive or an Antiderivative.
If f(x) is a function of x, the family of all antiderivatives of f(x) is called the integral of f(x) and is presented by $ \int{\text{f(x) }dx} $ .
The process of finding an integral of a given function f(x) is known as the integration of f(x).
The sign $ \int{\text{f(x) }dx} $ denotes integration of f(x).