Integration Question I have a midterm in 45 mins AH

[deerhake.11]

Integration Question.. I have a midterm in 45 mins! AH!

Homework Statement

integral with bounds [5,x^2] of f(t)dt = (5x^2)/(x^4+6) + C

f(t) = ?
C = ?

Homework Equations

Uhm.. I don't know. I am guessing you have to use the FTC.

The Attempt at a Solution

I've have tried so many different types of substitution and stuff with no luck. I really have no idea what method is required to solve this; I don't remember learning it.

 

[Schrodinger's Dog]

Probably way beyond me but have you tried partial fractions? And substitution?

Can you show what you've tried?

 

[quasar987]

The FTC says that

$$\int_a^b f(t)dt = F(b)-F(a)$$

where F'(t)=f(t).

So you want to find which function F(t) is such that F(x²) = -(5x^2)/(x^4+6). Then differentiate it to get f(t). About C, well C=F(5), so once you get F(t), finding C is a formality.

 

[HallsofIvy]

Yes, the derivative of [tex]\int_a^x f(t)dt[/itex] is f(x). But you are given $\int_a^{x^2} f(t)dt$. Let u= x² and use the chain rule: $\frac{df}{dx}= \frac{df}{du}\frac{du}{dx}$.

To determine C, let x= some convenient value and evaluate.