The integration of 1/(1+x^{2}) is tan^{-1}x+C. In this post, we will learn how to integrate 1 divided by 1 plus x square by the substitution method.

## Integration of 1/(1+x^{2}) Formula

1/(1+x^{2}) integral formula: The integration formula of 1/(1+x^{2}) is provided below.

∫ 1/(1+x^{2}) dx = tan^{-1}x+C.

## Integration of 1/(1+x^{2}) Proof

Now, we will prove that ∫1/(1+x^{2}) dx = tan^{-1}x+C. To prove that we will proceed as described below.

Let x=tanθ

Differentiating both sides using the fact d(tanθ)/dθ = sec^{2}θ, we get that

dx = sec^{2}θ dθ

Therefore, ∫1/(1+x^{2}) dx = ∫sec^{2}θ/(1+tan^{2}θ) dθ

= ∫sec^{2}θ/sec^{2}θ dθ using the trigonometric identity 1+tan^{2}θ=sec^{2}θ

= ∫dθ

= θ+C

= tan^{-1}x+C as x=tanθ.

So the integration of 1/(1+x^{2}) is tan^{-1}x+C where C is an integral constant. This is obtained by the substitution method where we put x=tanθ.

**Video Solution on Integration of 1/(1+x ^{2}) dx:**

**ALSO READ:**

**Integration of tan x** : The integration of tanx is sec^{2}x.

**Integration of cot x** : The integration of cotx is -cosec^{2}x.

**Integration of sec x** : The integration of secx is log_{e}|secx+tanx|.

**Integration of cosec x** : The integration of cosecx is log_{e}|cosecx-cotx|.

## FAQs

**Q1: What is the integration of 1/(1+x^2)**?

Answer: The integration of 1/(1+x^2) is equal to tan^{-1}x+C where C is an integration constant.