# Solve Integration by Parts: Arctan(4t)

• MillerGenuine
In summary, the conversation is about understanding the steps in solving the integral of arctan(4t). The solution involved integrating by parts and using the substitution rule. A 2 was multiplied inside and a 1/2 was put outside the integral to facilitate the substitution. The final answer involved substituting 4t back in for U and the 2u term disappeared due to the substitution rule.
MillerGenuine

## Homework Statement

$$\int arctan(4t)$$

## Homework Equations

I know what the answer is to the problem but when i look at the solution i have no idea how they get from one step to the next.

## The Attempt at a Solution

once we integrate by parts we get
$$1/4 U arctan(U) - 1/4 \int U/1+U^2$$ where U= 4t

from this they go to..

$$1/4 U arctan(U) - 1/8 \int 2U/1+U^2$$

clearly a 2 was multiplied inside and a 1/2 on the outside of the integral, but why? and how? because the next step shown is..

$$1/4 (4t arctan(4t) - 1/8 ln 16t^2 + c$$

which is the answer..i get that 4t was substituted back in for U, but i don't understand how and why the 2 was put in the integral and the 1/2 outside of it, and how in the last step the 2U somehow is gone?

MillerGenuine said:
once we integrate by parts we get
$$1/4 U arctan(U) - 1/4 \int U/1+U^2$$ where U= 4t

from this they go to..

$$1/4 U arctan(U) - 1/8 \int 2U/1+U^2$$

clearly a 2 was multiplied inside and a 1/2 on the outside of the integral, but why? and how? because the next step shown is..

You should use parantheses to make your work clearer. Anyway, note that (1+u2)' = 2u. The two is multiplied (and divided) so that you can apply the substitution rule for integration.

## 1. What is integration by parts?

Integration by parts is a method used in calculus to solve integrals of products of functions. It is based on the product rule of differentiation and involves breaking down the integral into two parts and applying a specific formula.

## 2. How do I know when to use integration by parts?

Integration by parts is typically used when the integral involves a product of two functions, with one of the functions being difficult to integrate. For example, in the case of arctan(4t), the integral involves a product of t and the arctan function, making it a good candidate for integration by parts.

## 3. How do I solve an integral using integration by parts?

To solve an integral using integration by parts, you first need to identify which part of the integral will be the "u" part and which will be the "dv" part. Then, use the formula "u dv = u v - ∫ v du" to solve for the integral, where u and v represent the respective parts.

## 4. What is the formula for integration by parts?

The formula for integration by parts is "u dv = u v - ∫ v du". This formula is derived from the product rule of differentiation and is used to solve integrals of products of functions.

## 5. Can I use integration by parts for any integral?

No, integration by parts is typically used for integrals that involve a product of two functions. It may not be useful for integrals that involve more complex functions or those that can be solved using other techniques, such as substitution or partial fractions.

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