Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

​Objectives

.I can find antiderivatives.

.I can use the Fundamental Theorem of Calculus.

Multiple Choice

The process of finding $$\frac{\text{d}y}{\text{d}x}=f'\left(x\right)$$  is known as_____

• Integration

• Differentiation

• Limitation

• continuation

Multiple Choice

The reverse process of finding $$y=f\left(x\right)$$  from $$\frac{\text{d}y}{\text{d}x}=f'\left(x\right)$$  is known as anti-Integration

• True

• False

Multiple Choice

$$\int f\left(x\right)dx\$$  is read as "the Indefinite Integral of f(x) with respect to x or Antiderivative"

• True

• False

​Antiderivative

​A function F is called the antiderivative of the function f on an interval I if F′ (x)=f(x) for every value of x in I.

​The expression F(x)+C is called the general antiderivative of f since it generalizes the family of antiderivatives of f.

​The Indefinite Integral

​The process of finding all antiderivatives of f(x) is called antidifferentiation. The other term used for this process is integration, and we also call an antiderivative an integral of f.

​The symbol ∫, called the integral sign, is used to denote the integral of a function. The integral of f(x) can be written as:

  ∫f(x)dx=F(x)+C

​This is called an indefinite integral.

​Basic Integration Rules

​the basic formula for integration can be derived from the differentiation formulas.

Let us introduce some of the basic formulas​

​The Indefinite Integral of a Constant

​Example:

Multiple Choice

What is the $$\int\frac{7}{3}dx$$  ?

• 7x + C

• 3x + C

• $$\frac{\text{3}}{\text{7}}x+C$$  

• $$\frac{\text{7}}{\text{3}}x+C$$  

​The Power Rule

Multiple Choice

What is the $$\int x^6dx?$$  

• 7x + C

• $$\frac{x^7}{7}$$  

• $$\frac{x^7}{7}+C$$  

• $$\frac{x^7}{7}-C$$  

Multiple Choice

What is the $$\int x^{\frac{2}{3}}dx?$$  

• $$\frac{3}{5}x\sqrt[3]{x^2}+C$$  

• $$\frac{5}{3}x\sqrt[3]{x^2}+C$$  

• $$\frac{5}{3}\sqrt[3]{x^5}+C$$  

• $$\frac{3}{5}\sqrt[3]{x^5}+C$$