**Integrating exponential and trigonometric functions**

In terms of derivation and integration, there are actually two types of exponential functions, *e*-based exponential functions and other exponential functions. The latter are much more difficult and come less often during high school.

Euler's number *e* is significant, precisely because the derivative of the function* f (x) = eΛ£* is the function* f(x)* itself. The same, of course, applies to integration

Often the exponent also has an inner function, so you have to be careful

Remember that the result of the integration should always be checked by derivation (or with a CAS calculator).

**Example 1**: Find the integral

Let's calculate another integral, where the base number is something other than *e*. Such calculations always go with the same idea

**Example 2**: Integrate

The integration of trigonometric functions is almost as easy, only the signs are easily confused.

**Derivative rules**

**Integration rules**

The tangent is still missing, we will return to it later. Again, one must be careful with the inner function and in area calculations knowing when the function is positive and when it is negative.

**Example 3**: Integrate

**Example 4**: Integrate

**Example 5**: Calculate the area bounded by the function *f(x) = sin (x) *and the x-axis between [0,2π
].

First we draw a graph

The zeros in the given range are 0, π , and 2π . Between [0, π ] the function gets only positive values, between [π , 2π ] only negative values. This allows the integral to be divided in half

Integral

Check with geogebra

**Turn on the subtitles if needed**