# Integrate x^3e^x^2 & xe^x/(x+1)^2

In summary: Dexter:In summary, I was unable to do the 2 integrals for the exercises, and I'm not sure how to do them by parts. I also don't know who Seneca is.
Hello everyone.
I have been trying to do those 2 exercices for a while now and I can't get it.. We just started doing Integration by parts (is that how you call it in english?)
here are the problems

1) $$\int{ (x^3)(e^{x^2})}$$
and
2) $$\int{ \frac{{(x)(e^x)}} { (x+1)^2}}$$

for the 2nd one I tried
u=e^(x^2) dv= x^3
du= 2xe^(x^2) v= (x^4)/4

From there I just get stuck..
For #2, I don't even know how to start it because there are 3 terms.. Can you do that by parts?
I know these are rookies question, but please explain me
thanks :)

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1) Try your luck with u = x² and dv = xe^x² (you can find v by substitution with x² = w)

2) Set

$$u = (x+1)^2 \ \ \ dv = xe^x dx \Rightarrow du = (2x^2 + 4x +2)dx \ \ \ v = \int(xe^x)dx$$

Let us find v by integrating by parts, using

$$w = x \ \ \ dz = e^x dx \Rightarrow dw = dx \ \ \ z = e^x \Rightarrow v = xe^x - \int e^x dx = xe^x - e^x$$

$$v = xe^x - e^x$$

Continue. You will have to do many other integrations by parts to get to the final answer.

The last one is tricky.

$$\int\frac{x}{(x+1)^{2}} \ dx=\int\frac{(x+1)-1}{(x+1)^{2}} \ dx=\int \frac{dx}{x+1} -\int \frac{d(x+1)}{(x+1)^{2}} =\ln (x+1)+\frac{1}{x+1}$$

Now

$$I:=\int e^{x}\frac{x}{(x+1)^{2}} \ dx$$

can be part integrated to get

$$I=e^{x}\left(\ln (x+1)+\frac{1}{x+1}\right)-\int e^{x}\left(\ln (x+1)+\frac{1}{x+1}\right) \ dx=e^{x}\left(\ln (x+1)+\frac{1}{x+1}\right)$$

$$-\int e^{x} \ln (x+1) \ dx-\int \frac{e^{x}}{x+1} \ dx$$

Use part integration for the first of the last 2 integrals

$$I=e^{x}\left(\ln (x+1)+\frac{1}{x+1}\right)-\left[e^{x}\ln(x+1)\right]+\int \frac{e^{x}}{x+1} \ dx-\int \frac{e^{x}}{x+1} \ dx =\frac{e^{x}}{x+1} + C$$

Daniel.

Last edited:
Dexter: What does "Docendo discitur" mean? Who's Seneca?

I'll let u discover who Seneca was.U can learn more from the internet.

"You learn by teaching others".It should be a motto for these forums...

Daniel.

## 1. What is the formula for integrating x^3e^x^2?

To integrate x^3e^x^2, we can use the substitution method. Let u = x^2, then du = 2x dx. This allows us to rewrite the integral as ∫ u*e^u/2 du, which can be solved using integration by parts.

## 2. How do you integrate xe^x/(x+1)^2?

To integrate xe^x/(x+1)^2, we can first use the substitution method by letting u = x+1, then du = dx. This allows us to rewrite the integral as ∫ (u-1)*e^u/ u^2 du, which can be solved using integration by parts.

## 3. Can these integrals be solved using any other methods?

Yes, in addition to the substitution and integration by parts methods, we can also solve these integrals using partial fractions. This method involves breaking the rational function into simpler fractions and then integrating each part individually.

## 4. Are there any special cases or restrictions to keep in mind while integrating these functions?

Yes, when using integration by parts, we need to make sure that the function we choose to differentiate does not become more complicated than the original function. This can lead to an infinite loop and make it difficult to solve the integral. Additionally, when using the substitution method, we need to choose the substitution carefully to avoid creating more complicated terms.

## 5. Can these integrals be solved using software or do they need to be solved by hand?

These integrals can be solved using software, such as a graphing calculator or a computer program. However, it is always beneficial to understand the concepts and methods behind integration in order to check the accuracy of the software and to be able to solve more complex integrals that may not have readily available solutions.

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