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Please help with Anti-derivitives! Need explanation and how to use them. Thank you

- Thread starter ATCG
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- #1

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Please help with Anti-derivitives! Need explanation and how to use them. Thank you

- #2

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Sounds like fun!

Antiderivatives might be used to solve problems like this:

What function has the derivative f'(x)=3x^{2}?

From your experience you might say f(x)=x^{3}.

That's a good guess, but you will notice that f(x)=x^{3} + 2 has the same derivative. Remember the derivative of a constant is zero. So this *differential equation*

f'(x)=3x^{2} actually has an infinite number of solutions. We represent this family of solutions by f(x)=x^{3}+C where C is an arbitrary constant. So now that you can find the antiderivative of 3x^{2}, how do you find antiderivatives in general?

Consider the differential equation dy/dx=x^{n} (n does not equal -1). What is the antiderivative of this equation - ie, what is y in terms of x? Consider

y=1/(n+1)*x^{n+1} - what is the derivative of y with respect to x? It is y'=x^{n}! We express this result in the following form.

dy/dx=x^{n}

dy=x^{n}dx .............multiply both sides by dx

[inte] dy= [inte] x^{n}dx ..........integrate both sides

y=1/(n+1)*x^{n+1} + C

The integral sign merely tells you to find the antiderivative of the equation. The "dx" and "dy" tell you what variable you are integrating (antidifferentiating) with respect to. To the left of the "dy" and "dx" is the derivative you are trying to undo.

So the left hand side [inte] dy = [inte] 1*dy means what function of y has a first derivative (taken wrt y) equal to 1? Obviously, it is f(y)=y since df/dy=dy/dy=1. The right hand side means what function of x has a derivative of x^{n}? This is the solution to the integral.

Another example,

dy/dx=cosx

dy=cosxdx

[inte] dy= [inte]cosxdx

y=sinx +C

Take the derivative of y to make sure I'm right.

Take a stab at this one:

dy/dx=sec^{2}x

What is y?

________

Technically, the left hand side should be y+C, but this is taken care of in the right hand side since C is entirely arbitrary.

Antiderivatives might be used to solve problems like this:

What function has the derivative f'(x)=3x

From your experience you might say f(x)=x

That's a good guess, but you will notice that f(x)=x

f'(x)=3x

Consider the differential equation dy/dx=x

y=1/(n+1)*x

dy/dx=x

dy=x

[inte] dy= [inte] x

y=1/(n+1)*x

The integral sign merely tells you to find the antiderivative of the equation. The "dx" and "dy" tell you what variable you are integrating (antidifferentiating) with respect to. To the left of the "dy" and "dx" is the derivative you are trying to undo.

So the left hand side [inte] dy = [inte] 1*dy means what function of y has a first derivative (taken wrt y) equal to 1? Obviously, it is f(y)=y since df/dy=dy/dy=1. The right hand side means what function of x has a derivative of x

Another example,

dy/dx=cosx

dy=cosxdx

[inte] dy= [inte]cosxdx

y=sinx +C

Take the derivative of y to make sure I'm right.

Take a stab at this one:

dy/dx=sec

What is y?

________

Technically, the left hand side should be y+C, but this is taken care of in the right hand side since C is entirely arbitrary.

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- #3

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Thanks alot StephenPrivitera!! I understand the anti-derivites now!

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- #5

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No problem. Happy to help anytime, anyday. Calculus is a very interesting topic.

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