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चरण द्वारा कदम

1. \int_{-\infty}^{\infty} \frac{1}{\left(x^{2} + 1\right)^{2}}\, dx

   First find an antiderivative F, then evaluate F(b) − F(a).

2. x = \tan{\left(\theta \right)}

   Trigonometric substitution.

3. \cos^{2}{\left(\theta \right)} = \frac{\cos{\left(2 \theta \right)}}{2} + \frac{1}{2}

   Rewrite the integrand into a friendlier form.

4. \int \frac{\cos{\left(2 \theta \right)}}{2} + \frac{1}{2}\, d_theta = \int \frac{\cos{\left(2 \theta \right)}}{2}\, d_theta + \int \frac{1}{2}\, d_theta

   The integral of a sum is the sum of the integrals.

5. \int \frac{\cos{\left(2 \theta \right)}}{2}\, d_theta = \frac{1}{2} \int \cos{\left(2 \theta \right)}\, d_theta

   Pull the constant \frac{1}{2} out of the integral.

6. u = 2 \theta,\quad du = 2\, d_theta

   Substitute u = 2 \theta.

7. \int \cos{\left(2 \theta \right)}\, d_theta = \int \frac{\cos{\left(u \right)}}{2}\, d_u

   Rewrite the integral in terms of u.

8. \int \frac{\cos{\left(u \right)}}{2}\, d_u = \frac{1}{2} \int \cos{\left(u \right)}\, d_u

   Pull the constant \frac{1}{2} out of the integral.

9. \int \cos{\left(u \right)}\, d_u = \sin{\left(u \right)}

   Standard trigonometric antiderivative.

10. = \frac{\sin{\left(2 \theta \right)}}{2}

    Substitute back u = 2 \theta.

11. \int \frac{1}{2}\, d_theta = \frac{\theta}{2}

    The integral of a constant c is c·x.

12. = \frac{x}{2 \left(x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(x \right)}}{2}

    Substitute back.

13. F(\infty) - F(-\infty) = \left(\text{NaN}\right) - \left(\text{NaN}\right)

    Fundamental theorem of calculus: plug in the limits.

14. = \frac{\pi}{2} \approx 1.5708

    Simplify.