Stuck on the integral: arctan (4t) dt

In summary, the conversation is about finding the integral of arctan(4t) dt and the confusion surrounding the derivative of arctan(x). The person is trying to use integration by parts, but is unsure about the integration rules and is confused about the derivative of arctan(4t). They have the answer to the problem, but are unsure how it was obtained and are seeking clarification.
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Homework Statement



The Integral is arctan (4t) dt


Homework Equations



I know how to do integration by parts, but I guess I have forgotten some of the integration rules.

The Attempt at a Solution



I set ∫arctan(4t)=u, and dt=dv

I know that the derivative of arctan(x) is 1/(1+x^2), But when I differentiate arctan(4t), it comes out as 4t/(1+16t^2). Why is this? To me it seems like it should be 1/(1+4t^2). I know how to do the rest, I have the answer, I'm just not sure how they got there. Thanks for any help.
 
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  • #2
dlikes said:
But when I differentiate arctan(4t), it comes out as 4t/(1+16t^2). Why is this?

Because you've calculated the derivative wrong. To tell you exactly where your error is, you'll need to post your steps.
 

FAQ: Stuck on the integral: arctan (4t) dt

1. What is the definition of an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value of a function over a given interval.

2. What does the arctan function represent?

The arctan function, also known as the inverse tangent function, is the inverse of the tangent function. It is used to find the angle in a right triangle given the length of the opposite and adjacent sides.

3. How do you solve an integral with the arctan function?

To solve an integral with the arctan function, you can use integration by parts or substitution. In this specific integral, you can use the substitution u = 4t to rewrite the integral as ∫arctan(u) du, which can then be solved using integration techniques.

4. What is the general formula for the integral of the arctan function?

The general formula for the integral of the arctan function is ∫arctan(x) dx = xarctan(x) - 1/2ln|1+x²| + C, where C is the constant of integration.

5. What are some real-life applications of the arctan function?

The arctan function has many applications in fields such as engineering, physics, and geometry. It is used to calculate angles in navigation and surveying, determine the direction of a magnetic field, and solve problems involving right triangles.

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