How to Integrate (1/a)arctan(u/a) + c Using Trigonometric Functions and Integration

(1/. a)arctan(u/. a)+c =

Apply basic properties of trigonometric functions and the integral

Apply basic properties of trigonometric functions and the integral.

The given expression is:
(1/a)arctan(u/a) + c

Let’s break it down step by step:

Step 1: Simplify the expression
We can rewrite the expression as follows:
arctan(u/a) / a + c

Step 2: Apply the integral rules
To integrate this expression, we will use the integral rule for the arctan function. The rule states that:
∫ dx / (1 + x^2) = arctan(x) + C

In our case, we have arctan(u/a), so we substitute u/a in place of x in the integral rule. Also, since we have 1/a in front of the arctan term, it can be pulled out as a constant. Therefore, the integral becomes:
(1/a) * ∫ du / (1 + (u/a)^2)

Step 3: Simplify the expression further
Simplifying the expression inside the integral:
(1/a) * ∫ du / (1 + (u^2/a^2))
= (1/a) * ∫ a^2 du / (a^2 + u^2)
= a / ∫ (a^2 + u^2) du

Step 4: Integrate the expression
Integrating the expression, we get:
a * arctan(u/a) + C1, where C1 is the constant of integration

Step 5: Incorporate the previously simplified expression
The final step is incorporating the constant C1. We can denote it as C, where C = a * C1.

Thus, the final answer is:
(1/a)arctan(u/a) + C (where C is a constant)

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