Presentation on theme: "Kale Shipley. This is the integration symbol. It indicates that the problem given must be integrated."— Presentation transcript:
This is the integration symbol. It indicates that the problem given must be integrated.
What is integration? Integration is the procedure of calculating an integral. An integral is a problem in which the derivative of an equation is given and the original equation must be found. This is why integration is also called antidifferentiation: because the opposite of differentiation is being performed.
What is an equation whose derivative is 2x? That would be. However, and also have derivatives of 2x. So, in order to show this capability of a constant, a “C” is added to the end of the answer. Thus, the answer to the above problem would be.
Indefinite Integration This is the formula for indefinite integration, which is what the problem from the previous example was. This means that once the answer is found, an unknown constant will be present (“C” in the previous example). F(x) stands for the function. Dx stands for the derivative.
This is the formula for definite integration. This means that an actual answer is going to be found, and no unknown constant will be present. Thus making the answer “definite.” To find the answer, you find the integral first, then plug the top number into the equation, then the bottom number, then subtract. So,.
There are six total rules for integration. 5 of them will be addressed in this powerpoint. These are: Power Rule U-Substitution Powers of Sine and Cosine Integration by Parts Partial Fractions
The power rule is the simplest rule in integration. It states that “u” is the function. “n” is the exponent.
u=x du=d x n=2 u=x du=d x n=1 u=x du=d x n=0
This is the second case that can be used in integration. It is used only when there are two functions and one of them isn’t the derivative of the other and cant be transformed to be.
How to do u-substitution First, take the most complicated function and set it equal to u. Solve for x. Take the derivative of each. Plug the various problems that have been found back in to the original equation, substituting u for x. Solve the integral in terms of u. Plug x back into the answer.
This is the third method of integration. As would be expected, it is used when sine and cosine are found in an integration problem. There are three different rules for integrating sine and cosine.
Take the odd power and break it up into a power of one and the remaining powers. For the trig that was rewritten, use its pythagorean identity. Use the distributive property and integrate using the power rule.
Take one of the odd powers and follow rule #1.
Either of the trig functions can be changed when solving the problem, if both functions exponents are three. When solving, find one answer. The other answer that could be found would have opposite signs (+ and -) and opposite functions.
Change both functions to their half-angle formulas. Repeat as many times as needed. Integrate using the power rule.
This is the fourth method of integration. This method is used when there are two completely unrelated functions in the integral. This includes combinations of x, lnx, e^x, and sinx or cosx.
In the equation, the derivative of u must be able to be taken. Also, the integral of dv must be able to be taken. In most cases, you want u to be the polynomial if possible.
This is the fifth and final method of integration that we have learned. This method is used when the function in the integral is a rational polynomial in which the denominator is factorable. It is much easier to show how to do this method than it is to describe instructions. So, pay attention to the following example.
Factor the denominator and put into partial fractions with unknown constants in numerator Coefficient of linear function in numerator Constant in numerator Plug back in to integral Solve for the unknow n constan ts Multiply this by this to get this 1)1) 2) 3)