# How do you evaluate the integral #int dx/(x^3+x)#?

We can write the integral as:

now we can substitute:

so we have:

and using the trigonometric identity:

so that finally:

By signing up, you agree to our Terms of Service and Privacy Policy

To evaluate the integral ∫ dx/(x^3 + x), we can start by factoring the denominator:

x^3 + x = x(x^2 + 1)

Now, we can use partial fraction decomposition to express the integrand as the sum of simpler fractions. Since the denominator has a linear term and a quadratic term, we write:

1/(x^3 + x) = A/x + (Bx + C)/(x^2 + 1)

Multiplying both sides by (x^3 + x) to clear the denominator and then solving for A, B, and C, we find:

A = 1, B = -1/2, C = 1/2

Now, we can rewrite the integral as:

∫ dx/(x^3 + x) = ∫ (1/x - (1/2)*(x/(x^2 + 1)) + (1/2)*(1/(x^2 + 1))) dx

Now, we integrate each term separately:

∫ (1/x) dx = ln|x| ∫ (x/(x^2 + 1)) dx = (1/2) * ln|x^2 + 1| ∫ (1/(x^2 + 1)) dx = arctan(x)

Putting it all together, we have:

∫ dx/(x^3 + x) = ln|x| - (1/2) * ln|x^2 + 1| + (1/2) * arctan(x) + C

where C is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7