## ∫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.

##### More Answers:

Step-by-Step Guide | Finding the Integral of cot(x) with Respect to x Using SubstitutionEvaluate the Integral of csc(x) Using Trigonometric Identity and Substitution

How to Solve the Integral ∫csc(x)cot(x)dx using U-Substitution Technique