How do I integrate a definite integral with variable limits and constants?

[lauriecherie]

Homework Statement

can someone please explain to me how to intergrate this:

the definite integral from 0 to PI of (.25*PI) (1+kh) g (h + .2) dh

I can leave g, PI, and k in the formula.

Homework Equations

The Attempt at a Solution

 

[slider142]

What have you attempted? Do you know how to integrate polynomials?

 

[lauriecherie]

I'm not sure what I need to do. The k is what's causing me problems. I tried factoring but that got me no where. Can you explain what I need to do or explain to me how the book got this answer to this integral? This one of course isn't my problem but can you explain how they intergrated this one? It is similar to the one I need to solve.


The definite integral from 0 to 1.5 of (.25 * PI) (1 + kh) g (h + 0.3) dh

The answer: .366 (k + 1.077) gPI

 

[CFDFEAGURU]

Integrate the function with respect to h and then insert the upper value of the integral for h and then do the same for the lower value of the integral and subtract the two.

 

[CFDFEAGURU]

Remember that constants come outside the integral.

 

[lauriecherie]

Ok so I can pull out PI, g, and .25? Or just PI and g?

And do i distribute (1 + kh) * (h + .2)?

 

[slider142]

Is k a constant (A number which does not depend on the variable of integration h)? If so, remember this rule of integration of two integrable functions f and g:
$$\int (f + g) = \int f + \int g$$
and if k is a constant:
$$\int kf = k\int f$$
Also, remember the power rule of differentiation:
$$\frac{d}{dx}(x^n) = nx^{n-1}$$
from which we get:
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$
where C is an arbitrary constant of integration.
The best course of action is then to multiply everything out so that you are left with a polynomial, where you can integrate each term in the sum separately.

 

[lauriecherie]

So I'm getting...

.25*PI*g * (The definite integral from 0 to 1.5 of (h + kh^2 + .2 + .2kh).

I don't know what to do with this k.

If I integrate I'm getting

.25*PI*g * { (h^2/2) + ((kh^3)/(3)) + (.2h) + (.1k*h^2)} evaluated from 0 to 1.5

 

[Redbelly98]

Looks good, now just evaluate it with the limits 0 to 1.5.