Mathematics

# Evaluate: $\displaystyle\int \dfrac{1}{x(x^2+1)}dx$.

##### SOLUTION
We are given,
$I=\int \dfrac{dx}{x (x^{2}+1)}$
Let $x= \tan \theta$
$dx= \sec^{2} \theta d \theta$  (differentiation)
$dx = (1+ \tan^{2} \theta ) d \theta$
$\dfrac{dx}{ (1+x^{2})}= d \theta$
$I =\int \dfrac{d \theta }{ \tan \theta}$
$= \int \cot \theta d \theta$
$I= \int \dfrac{\cos \theta }{\sin \theta } d \theta$
Let $\sin \theta = t$
$\cos d\theta= dt$ (differentiation)
$I = \int \dfrac{dt}{t}$
$I =In |t|+ C$
$I= In |\sin \theta |+ C$
Now, $\sin \theta = (\tan \theta \div \sec \theta)$
$= x / \sqrt{1+x^{2}}$
$I= In \left( \dfrac{x}{\sqrt{1+x^{2}}} \right)+ C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

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