# Integration By Parts(stuck on one part)

• rey242
In summary, the definite integral Integral[0,1] Ln(x^2+1) dx can be evaluated by first dividing the polynomial x^2+1 into x^2 and expressing it as a quotient and remainder. This allows for the integral to be rewritten as x* Ln(x^2+1)- Integral [0,1] (2x^2/(x^2+1) dx. The second integral can then be solved by using the formula 1/(1+x^2) = 1-1/(1+x^2), which can be integrated as an inverse trigonometric function. This method can be used to evaluate the original definite integral.
rey242

## Homework Statement

Evaluate the Definite Integral
Integral[0,1] Ln(x^2+1) dx

## The Attempt at a Solution

So far I have done the first part,
U= Ln (x^2+1) Du= 2x/(x^2+1)dx
Dv= 1 dx V=x

x* Ln(x^2+1)- Integral [0,1] (2x^2/(x^2+1) dx

But I'm stuck on the second integral, I have tried punching into a TI-89, but It comes out to a weird answer. Can someone help?

Start by dividing the polynomial x^2+1 into x^2 to express it as quotient and remainder.

What do I do after that? How does that help?

x^2/(1+x^2)=1-1/(1+x^2). You can integrate 1 and you should be able to integrate 1/(1+x^2). It's an inverse trig function.

Oh wow, thanks!

## 1. What is Integration by Parts?

Integration by Parts is a mathematical method used to evaluate integrals that involve products of functions. It allows us to transform a complex integral into a simpler one that can be easily solved.

## 2. How do I know when to use Integration by Parts?

You can use Integration by Parts when you have an integral that involves a product of two functions, and you are unable to directly evaluate it using other methods such as substitution or trigonometric identities.

## 3. What is the formula for Integration by Parts?

The formula for Integration by Parts is ∫udv = uv - ∫vdu, where u and v are the two functions being multiplied together, and du and dv are their respective differentials.

## 4. How do I choose u and dv in the Integration by Parts formula?

When using the Integration by Parts formula, you should choose u and dv in a way that simplifies the integral on the right side of the equation. A general rule is to choose u as the part of the integrand that becomes simpler when differentiated, and dv as the part that becomes simpler when integrated.

## 5. What should I do if I get stuck on one part of the Integration by Parts process?

If you get stuck on one part of the Integration by Parts process, you can try using other integration techniques like substitution or partial fractions. You can also try manipulating the integrand algebraically to make it easier to integrate. If you are still unable to solve the integral, it may be helpful to consult a calculus textbook or seek assistance from a math tutor or professor.

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