Question

Integrate the following
$\int \frac{2x}{1 + {x}^{2} } dx$
​

Answer 1

Answer:

$\displaystyle\sf\int \frac{2x}{1 + {x}^{2} } \,dx = ln|x^2 + 1| + c$

Step-by-step explanation:

We are asked to find the anti derivative of given function.

$\displaystyle\sf\int \frac{2x}{1 + {x}^{2} } dx$

We will solve this question using the method of substitution.

$\rm Let \: {x}^{2} + 1 =t \implies2x \: dx = dt$

Therefore the given integral changes to,

$\displaystyle \implies\sf\int \frac{1}{t}dt$

Here, we can use the below identity:

• $\boxed{ \sf\int \frac{dx}{x} = \ln |x| + c }$

By using it, we get:

$\displaystyle \implies\sf \ln |t| + c$

Substituting back the value of t.

$\boxed{ \displaystyle \implies\sf \ln | {x}^{2} + 1 | + c}$

This is the required answer.

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