Step-by-Step Solution | Finding the Integral of 1/(x²+a²)

∫1/(x²+a²)dx

The given integral can be written as ∫(1/(x²+a²))dx

The given integral can be written as ∫(1/(x²+a²))dx. Let’s solve it step by step.

1. To start, we can rewrite the integrand as 1/((√a)²+x²). This rearrangement helps us recognize the integral as the inverse tangent function.

2. Now, let’s make a substitution:
Let x = √a * tan(theta), where theta is a new variable.
Note that dx = √a * sec²(theta) d(theta).

3. Substituting the values of x and dx in terms of theta, we have:
∫(1/(x²+a²))dx = ∫(1/((√a)²+x²))dx
= ∫(1/((√a)²+(√a * tan(theta))²)) * (√a * sec²(theta)) d(theta)
= ∫(1/(a + a * tan²(theta))) * (√a * sec²(theta)) d(theta)
= ∫(1/(a * sec²(theta))) * (√a * sec²(theta)) d(theta)

4. Simplifying the expression further, we cancel out the sec²(theta) terms to obtain:
∫(1/(a * sec²(theta))) * (√a * sec²(theta)) d(theta) = ∫(1/√a) d(theta)

5. Since 1/√a is a constant, we can take it out of the integral:
∫(1/√a) d(theta) = (1/√a) ∫d(theta)

6. Integrating d(theta) with respect to theta just gives theta:
(1/√a) ∫d(theta) = (1/√a) * theta

7. However, we still have to find the value of theta. So let’s go back to our substitution x = √a * tan(theta). Solving for theta, we get:
theta = arctan(x/√a)

8. Therefore, the final result is:
∫(1/(x²+a²))dx = (1/√a) * theta = (1/√a) * arctan(x/√a) + C

C is the constant of integration. So the integral of 1/(x²+a²)dx is (1/√a) * arctan(x/√a) + C, where C is a constant.

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