indefinite integral I

Calculus Level 3

∫ 1 ( x 2 + 1 ) 2 d x \large \int \frac{1}{(x^2+1)^2}dx

Evaluate the indefinite integral above.

Notation: C C denotes the arbitrary constant of integration.

Use

  • cos ⁡ 2 x = 1 + cos ⁡ 2 x 2 \cos^2 x = \dfrac {1+\cos2x}{2}
  • sin ⁡ ( arctan ⁡ x ) = x x 2 + 1 \sin(\arctan x) = \dfrac{x}{\sqrt{x^2+1}}
  • cos ⁡ ( arctan ⁡ x ) = 1 x 2 + 1 \cos(\arctan x) = \dfrac{1}{\sqrt{x^2+1}}
1 2 arctan ⁡ x + x 2 ( x 2 + 1 ) \frac{1}{2} \arctan x+\frac{x}{2(x^2+1)} 1 3 arctan ⁡ x + 1 2 ( x 2 + 1 ) + C \frac{1}{3} \arctan x+\frac{1}{2(x^2+1)}+C 1 2 arctan ⁡ x + x 2 ( x 2 + 1 ) + C \frac{1}{2}\arctan x+\frac{x}{2(x^2+1)}+C

Dimitris Kouroupos by dimitris kouroupos , 23, Greece

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Chew-Seong Cheong
Apr 28, 2017

I = ∫ 1 ( x 2 + 1 ) 2 d x Let x = tan ⁡ θ , d x = sec ⁡ 2 θ d θ = ∫ sec ⁡ 2 θ ( tan ⁡ 2 θ + 1 ) 2 d θ = ∫ sec ⁡ 2 θ sec ⁡ 4 θ d θ = ∫ 1 sec ⁡ 2 θ d θ = ∫ cos ⁡ 2 θ d θ = ∫ 1 + cos ⁡ 2 θ 2 d θ = θ 2 + sin ⁡ 2 θ 4 + C where C is the constant of integration. = 1 2 θ + sin ⁡ θ cos ⁡ θ 2 + C = 1 2 arctan ⁡ x + x 2 ( x 2 + 1 ) + C \begin{aligned} I & = \int \frac 1{(x^2+1)^2} dx & \small \color{#3D99F6} \text{Let }x = \tan \theta, \ dx = \sec^2 \theta \ d\theta \\ & = \int \frac {\sec^2 \theta}{\left(\tan^2 \theta + 1\right)^2} d\theta \\ & = \int \frac {\sec^2 \theta}{\sec^4 \theta} d\theta \\ & = \int \frac 1{\sec^2 \theta} d\theta \\ & = \int \cos^2 \theta \ d\theta \\ & = \int \frac {1+\cos 2\theta}2 \ d\theta \\ & = \frac \theta 2 + \frac {\sin 2\theta}4 + \color{#3D99F6} C & \small \color{#3D99F6} \text{where } C \text{ is the constant of integration.} \\ & = \frac 12 \theta + \frac {\sin \theta \cos \theta}2 + C \\ & = \boxed{\dfrac 12 \arctan x + \dfrac x{2(x^2+1)}+C} \end{aligned}

9 Helpful 0 Interesting 0 Brilliant 0 Confused

i got it wrong only because I did not notice the c damn it!

Raven Herd - 3 years, 9 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...