# How do solve this integral

1. Jun 8, 2010

### Taturana

1. The problem statement, all variables and given/known data

I need to proceed to solve this integral:

2. Relevant equations

$$\int_{0}^{a}\frac{s \ ds}{\sqrt{y^{2} + s^{2}}} = \left [ \sqrt{y^{2} + s^{2}} \right ]_{s=0}^{s=a}$$

3. The attempt at a solution

I don't understand what he did from the left-hand side of the equation to the right-hand side... could someone explain me, please?

Thank you,
Rafael Andreatta

2. Jun 8, 2010

### Staff: Mentor

It seems to me that an ordinary substitution was used: u = y2 + s2, du = 2s ds.

3. Jun 8, 2010

### Taturana

I tried some math here and I still don't understand...

When you get an integral of that type, how do you know when you need to make this kind of substitution?

Could you explain me more clearly?

Thank you very much...

4. Jun 8, 2010

### rock.freak667

u=y2+s2

if y is a constant and you differentiate wrt s you will get

du/ds = 2s or du=2s ds, meaning that s ds = du/2

It comes with practice really. If you have this integral

$$\int \frac{x^2}{x^3+1} dx$$

you notice that d/dx(x3+1) = 3x2

so if you used a substitution of t=x3+1, dt = 3x2 dx or dt/3 = x2 dx.

So in the integral, you can replace x2 dx with dt/3.

5. Jun 8, 2010

### Staff: Mentor

Show us what you're trying and we'll straighten you out.

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