Solving Basic Integrals: A Tutorial

[lemonlee]

Homework Statement

I've been trying to find a website to explain this to me, but I'm still confused as to how to do an integral. The one that I have been trying to figure out is this:

Integral from 0 to d of F(x)dx

i can't figure out where the integral sign is either...sorry! but 0 is on the bottom and d on the top

Homework Equations

I know that it's almost the opposite of deriving an equation, d/dx

The Attempt at a Solution

no idea! I need to figure this out for a quiz on thursday and my professor just confuses me more when i try to get help.

 

[Mark44]

The operations of differentiation and integration (or antidifferentiation) are pretty much inverse operations.

If you know, for example that
$$\frac{d}{dx} x^3 = 3x^2$$ then you know an antidifferentiation formula,
$$\int{ 3x^2 dx} = x^3 + C$$

or equivalently,
$$\int{ x^2 dx} = \frac{1}{3}x^3 + C$$

If you're working with a definite integral (with limits of integration), you still need to find an antiderivative, as you do with an indefinite integral like the ones above in my post. The only difference is that you evaluate the antiderivative at upper limit and lower limit and then subtract the latter from the former.


One thing you said, "opposite of deriving an equation, d/dx" is incorrect. You don't apply the d/dx operator to an equation; you apply it to a function.

 

[lemonlee]

so if the equation that I'm trying to intergrate is F=-kx^4
i first have to find the derivative of -kx^4=4x^3 (right?) and then

4x^3dx=1/4x4+C

and from there i put in the two values (d and 0) and subtract them to find the definite integral?

 

[HallsofIvy]

NO, because the derivative of (1/4)x⁴+ C is (1/4)(4 x³)= x³, not 4x³. You need to multiply your result by -4: $\int -k x^3 dx= -k \int x^3dx= -4k x^4+ C$.

 

[Mark44]

so if the equation that I'm trying to intergrate is F=-kx^4
i first have to find the derivative of -kx^4=4x^3 (right?) and then

4x^3dx=1/4x4+C

and from there i put in the two values (d and 0) and subtract them to find the definite integral?

No, what you have to do is find a function whose derivative is -kx^4. It will be some multiple of x^5.

 

[Mark44]

NO, because the derivative of (1/4)x⁴+ C is (1/4)(4 x³= x³, not 4x³. You need to multiply your result by -4: $\int -k x^4 dx= -k \int x^4dx= -4k x^4+ C$.

Halls, the last expression on the right is incorrect. It should be $$-k \frac {x^5}{5} + C$$


lemonlee,
The check for the above is to differentiate the result gotten by antidifferentiation.

$$\frac{d}{dx}(-k \frac {x^5}{5} + C )= -k \frac{5x^4}{5} + 0 = -kx^4$$

 

[HallsofIvy]

Right- his answer had me integrating the wrong thing!