In this lesson, we will look at Basic Techniques of Integration. These techniques are important for anyone learning calculus.

Integration is like the opposite of differentiation, and knowing the basic rules can help you solve tougher calculus problems later on.

Basic Integration Rules

1. Power Rule: This is one of the most common rules. It says that if you have any real number ( n ) (except for -1), you can integrate it like this:

   $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C,$$

   Here, ( C ) is a constant number that you add to your answer. This rule makes it easy to integrate polynomial functions.

2. Constant Rule: This rule is simple. If ( k ) is a constant number, then:

   $$\int k \, dx = kx + C.$$

   This shows that if you have a constant multiplied by a function, you can take it out of the integral.

3. Sum/Difference Rule: This rule tells us that when we add or subtract functions, we can integrate them separately. So,

   $$\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx,$$

   and

   $$\int (f(x) - g(x)) \, dx = \int f(x) \, dx - \int g(x) \, dx.$$

   This helps when we have more complicated functions to integrate.

Common Integrals

Using the rules helps, but some integrals show up a lot. Here are a few important ones to memorize:

• Exponential Function:

  $$\int e^x \, dx = e^x + C.$$

• Sine Function:

  $$\int \sin(x) \, dx = -\cos(x) + C.$$

• Cosine Function:

  $$\int \cos(x) \, dx = \sin(x) + C.$$

Knowing these common integrals can really make solving calculus problems easier.

Connection to Differentiation

To really understand integration, it's important to know how it relates to differentiation. This is captured in the Fundamental Theorem of Calculus. This theorem shows that differentiation and integration are reverse processes:

1. If ( F(x) ) is an antiderivative of ( f(x) ), which means:

   $$F'(x) = f(x),$$

   then:

   $$\int f(x) \, dx = F(x) + C.$$

2. On the other hand, if we want to find the definite integral of ( f(x) ) from ( a ) to ( b ), we can do it like this:

   $$\int_{a}^{b} f(x) \, dx = F(b) - F(a).$$

   This shows how integration can measure the total of ( f(x) ).

Understanding how integration and differentiation work together is key for learning calculus. Mastering these basic integration techniques will make it much easier for you to tackle calculus problems!