Question

What is the value of integral $\int{\dfrac{1}{1+4{{x}^{2}}}}dx$?

Answer 1

Hint: For solving this question you should know about the integration of functions. In this we will use the trigonometric identity and then convert it in the form of ${{\sec }^{2}}x$ and then we substitute this value in the integral and by solving it we will get the answer for the integral.

Complete step-by-step solution:
According to our question we have to find the value of $\int{\dfrac{1}{1+4{{x}^{2}}}}dx$. So, for solving the value of this question we will use the trigonometric identities and we know that the commonly used trigonometric identity is ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ and since our function is in the form of $\int{\dfrac{1}{1+4{{x}^{2}}}}dx$, we have to convert this identity to another identity. By dividing with ${{\cos }^{2}}\theta $ on both sides, we will get,
$\dfrac{{{\sin }^{2}}\theta }{{{\cos }^{2}}\theta }+\dfrac{{{\cos }^{2}}\theta }{{{\cos }^{2}}\theta }=\dfrac{1}{{{\cos }^{2}}\theta }$
Since we know that,
$\dfrac{{{\sin }^{2}}\theta }{{{\cos }^{2}}\theta }={{\tan }^{2}}\theta $
So, we will get it as,
${{\tan }^{2}}\theta +1={{\sec }^{2}}\theta $
For integration, we can try the substitution as $2x=\tan \left( u \right)$. From this we get,
$2dx={{\sec }^{2}}\left( u \right).du$
Now putting this in the integral, we will get as follows,
$\int{\dfrac{1}{1+4{{x}^{2}}}}dx=\dfrac{1}{2}\int{\dfrac{{{\sec }^{2}}\left( u \right)}{1+{{\tan }^{2}}\left( u \right)}}du$
Now using the trigonometric identity on the denominator of the fraction, we will get as,
$\dfrac{1}{2}\int{\dfrac{{{\sec }^{2}}\left( u \right)}{{{\sec }^{2}}\left( u \right)}}du=\dfrac{1}{2}\int{1.du}$
So, by integrating the substitution gives us,
$\begin{align}
  & =\dfrac{1}{2}u+c \\
 & \because u={{\tan }^{-1}}2x \\
 & =\dfrac{1}{2}{{\tan }^{-1}}\left( 2x \right)+c \\
\end{align}$
So, the value of $\int{\dfrac{1}{1+4{{x}^{2}}}}dx$ is $\dfrac{1}{2}{{\tan }^{-1}}\left( 2x \right)+c$.

Note: For solving the integration of the functions like this always use the trigonometric identity because if you will not use the trigonometric identities then you can’t solve it even one step forward because this is the right way to solve this type of question. And only this method can solve the integral of these types.