Integration of 1/(1+x^2) | How to Integrate 1/(1+x^2)

The integration of 1/(1+x2) is tan-1x+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+x2) Formula

1/(1+x2) integral formula: The integration formula of 1/(1+x2) is provided below.

∫ 1/(1+x2) dx = tan-1x+C.

Integration of 1/(1+x2) Proof

Now, we will prove that ∫1/(1+x2) dx = tan-1x+C. To prove that we will proceed as described below.

Let x=tanθ

Differentiating both sides using the fact d(tanθ)/dθ = sec2θ, we get that

dx = sec2θ dθ

Therefore, ∫1/(1+x2) dx = ∫sec2θ/(1+tan2θ) dθ

= ∫sec2θ/sec2θ dθ using the trigonometric identity 1+tan2θ=sec2θ

= ∫dθ

= θ+C

= tan-1x+C as x=tanθ.

So the integration of 1/(1+x2) is tan-1x+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+x2) dx:

Integration of tan x : The integration of tanx is sec2x.

Integration of cot x : The integration of cotx is -cosec2x.

Integration of sec x : The integration of secx is loge|secx+tanx|.

Integration of cosec x : The integration of cosecx is loge|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.