Indefinite Integral (Antiderivative) - an integral expressed without limits, and so containing an arbitrary constant
The formula is: F(x)=∫f(x)dx- where F(x) equals the integral of f(x)
The integral of a function is going from the derivative, back to the original function
The formula is: F(x)=∫f(x)dx- where F(x) equals the integral of f(x)
The integral of a function is going from the derivative, back to the original function
Indefinite integrals, or antiderivatives, use the opposite process of differentiation (taking derivatives).
∫ is the symbol for the sum of an infinite number of infinitely small areas (or other variables).
Since the antiderivative does not include constants, a +C (arbitrary constant) must be added to each antiderivative. The C holds the place for any constant, such as 1 or 667 (electrical Engineers use +K instead of +C).
∫ is the symbol for the sum of an infinite number of infinitely small areas (or other variables).
Since the antiderivative does not include constants, a +C (arbitrary constant) must be added to each antiderivative. The C holds the place for any constant, such as 1 or 667 (electrical Engineers use +K instead of +C).
Video explanation of example 1:
https://www.educreations.com/lesson/view/example-1-antiderivative/42282960/?s=hey02e&ref=link
https://www.educreations.com/lesson/view/example-1-antiderivative/42282960/?s=hey02e&ref=link
Integration By Substitution
Ex 1)
- To integrate functions that have mutipule parts such as the one above. You must use what is known as integration by substitution of U substitution.
- This form of integration has all the same steps as normal integration with one extra part. Whenever a function has an inside where the derivative of that piece is not 1, then you have to use U substitution.
- You must first set a u for the above example u=x+1 (you do not necessarily have to use U substitution for this problem because the inside piece's derivative is equal to one)
- Then you must take the derivative of this equation leaving you with du=1 dx, so we would substitute du for dx as they did and have the integral of u^2
- Then you integrate u like a normal integral, but after integrating you must plug what you set u equal to in the beginning back into the equation to get the answer
- Every often du does not directly equal dx and do you must manipulate them to be equal to the rest of the equation like in the following example
- Because the derivative of u was not exactly what was left in the equation. You must manipulate du so that it equal to what is left in the equation.
- This made it 1/4 du and so after integration the integral must be multiplied by 1/4
Set A
Set B
Solutions
A)
B)
Using Antiderivatives to find position, velocity, and acceleration
If S(t)=position, S’(t)=v(t), S’’(t)=v’(t)=a(t),
If S(t)=position, S’(t)=v(t), S’’(t)=v’(t)=a(t),
These antiderivative functions can be used to work backwards when given an acceleration equation to find the velocity or position
Ex. A particle moves in a straight line and has an acceleration of a(t)= 8t-6. v(0)= 2 m/s and s(0)= 5 m/s. Find the position equation.
GRAPHING tHE ANTIDERIVATIVE BY HAND
Sources
Bourne, Murray. “Antiderivatives and the Indefinite Integral.” Interactive
Mathematics. Interactive Mathematics, 13 June 2016. Web. Retrieved 2 Nov 2016.
Dawkins, Paul. “Calculus-1 Notes.” Paul’s Online Math Notes. Paul Dawkins
2016. Web. Retrieved 2 Nov 2016.
Finney, Ross L, et al. Calculus Graphical, Numerical, Algebraic. Prentice Hall, 2003.
Antiderivatives-Examples, Part 4- Position, Velocity and Acceleration. Math Easy Solutions, 6 Oct 2012, Web. 17 Nov 2016.
Bourne, Murray. “Antiderivatives and the Indefinite Integral.” Interactive
Mathematics. Interactive Mathematics, 13 June 2016. Web. Retrieved 2 Nov 2016.
Dawkins, Paul. “Calculus-1 Notes.” Paul’s Online Math Notes. Paul Dawkins
2016. Web. Retrieved 2 Nov 2016.
Finney, Ross L, et al. Calculus Graphical, Numerical, Algebraic. Prentice Hall, 2003.
Antiderivatives-Examples, Part 4- Position, Velocity and Acceleration. Math Easy Solutions, 6 Oct 2012, Web. 17 Nov 2016.