1. ## Integration by substitution

Hello, I have to integrate the following by substitution, but I am having trouble... any help would be appreciated...

$\int x \log (x^2 + 1)dx$. thanks

2. Originally Posted by slevvio
Hello, I have to integrate the following by substitution, but I am having trouble... any help would be appreciated...

$\int x \log (x^2 + 1)dx$. thanks
i suppose by log here, you mean ln. just make the substitution u = ln(x^2 + 1)

3. Thanks for the response however using that substitution I end up with

$\frac{1}{2}\int \ln(u) du$, which I don't know how to integrate. Thanks

4. Originally Posted by slevvio
Thanks for the response however using that substitution I end up with

$\frac{1}{2}\int \ln(u) du$, which I don't know how to integrate. Thanks
That's what happens if you use the substitution: $u=x^2 + 1$

5. Originally Posted by slevvio
Thanks for the response however using that substitution I end up with

$\frac{1}{2}\int \ln(u) du$, which I don't know how to integrate. Thanks
see here

i'm not sure that's the integral we would get though...let me see. again, i assume you mean ln when you type log

6. Originally Posted by colby2152
That's what happens if you use the substitution: $u=x^2 + 1$
correct, which happens to be easier than my substitution

well, not really, you'd end up with $\frac 12 \int ue^u~du$ for mine, which is almost the same thing, in terms of difficultly

7. hehe i knew I was doing somethign right... thanks a lot, our maths lecturer at uni is German and she uses Log when she means ln in the notes i just copied it straight out

8. Hello, slevvio!

I have to integrate the following by substitution. .?

. . . $\int x \log (x^2 + 1)dx$
This cannot be integrated by substitution alone.

We must integrate by parts . . .

. . $\begin{array}{ccccccc}u & = & \ln(x^2+1) & \quad & dv & = & x\,dx \\ du & = & \frac{2x}{x^2+1}\,dx & \quad & v & = & \frac{1}{2}x^2 \end{array}$

Then we have: . $\frac{1}{2}x^2\ln(x^2+1) - 2\int\frac{x^3}{x^2+1}\,dx$

. . . $=\;\frac{1}{2}x^2\ln(x^2+1) - 2\int\left(x - \frac{x}{x^2+1}\right)\,dx$ . . . . etc.