# Integration by Parts: Solving Integrals with √(1+x^2) and x

• Chris Fernandes
In summary, the conversation discusses the use of the fundamental theorem of calculus to solve an integration problem involving the functions f(x) = √(1+x^2) and g(x) = x. The integration is simplified using substitution and the result is compared to the original functions. Further manipulation may lead to a solution.

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## Homework Equations

∫ f(x) g'(x) dx = f(x) g(x) - ∫ f '(x) g(x) dx

f(x)=√(1+x^2)
f '(x)=x * 1/√(1+x^2)

g'(x)=1
g(x)=x

## The Attempt at a Solution

∫ √(1+x^2) * 1 dx
=x * √(1+x^2) - ∫ x^2 * 1/√(1+x^2) dx

Further integration just makes the result look further from what it's supposed to look like

Chris Fernandes said:
=x * √(1+x^2) - ∫ x^2 * 1/√(1+x^2) dx
The latter part may help to go to your answer. You maybe can try to make it become ##\frac{x^2}{\sqrt{1+x^2}}=\sqrt{1+x^2}-\frac{1}{\sqrt{1+x^2}}.##

tommyxu3 said:
The latter part may help to go to your answer. You maybe can try to make it become ##\frac{x^2}{\sqrt{1+x^2}}=\sqrt{1+x^2}-\frac{1}{\sqrt{1+x^2}}.##

Thank you!

## 1. What is integration by parts?

Integration by parts is a method for solving integrals that involves breaking down the integral into two parts and applying the product rule for derivatives.

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

You should use integration by parts when the integral contains a product of functions or when it can be rewritten as a product of functions.

## 3. How do I solve integrals with square roots and x using integration by parts?

To solve integrals with square roots and x using integration by parts, you need to choose which function will be u and which will be dv in the integration by parts formula. Then, you can apply the formula and simplify the resulting integral until it can be solved.

## 4. Can integration by parts be used for all types of integrals?

No, integration by parts can only be used for certain types of integrals, specifically those that can be rewritten as a product of functions or those that contain a product of functions.

## 5. Are there any tips for using integration by parts effectively?

Yes, some tips for using integration by parts effectively include choosing u and dv strategically, considering using integration by parts multiple times, and simplifying the resulting integral as much as possible before attempting to solve it.