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
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