**Integrating quotients**

As a rule, the quotients are integrated by the integral formula of the power function. Keep in mind that they are often divided into two branches, such as the previously calculated integral:

Such quotient integrals can be calculated by known methods, in all other cases, except when the denominator has the first power. We remember from the derivation that the derivative of the natural logarithm gives the function *1/x*, so it follows

The absolute value signs within the logarithm ensure that the result is defined in the same domain as the function to be integrated (i.e.,* x* gets all values except the value 0).

If there is a function denominator instead of just *x*, the derivative of its internal function must again be found inside the integral. Otherwise, the calculation formula is the same

The zeros of the function must be removed from the definition area. For this reason, we have a piecewise-defined function that we need our own integration constants for different pieces.

The simplest example of this is

The zero of the denominator divides the function into two branches

The integral is divided into two parts on different sides of the zero point

In the latter part, the absolute value signs can be forgotten because the function *2x + 1* is positive in this range.

**Example 1**: Integrate

Again, in the latter part, the function within the logarithm is positive (the exponential function is strictly increasing), so the absolute value signs can be left out.

**Example 2**: Let *x* be between

Find the integral of the tangent

In a given range, cos (x) is positive, so the denominator has no zeros. Furthermore, it is known that -sin (x) is a derivative of cosine so

If there were zeros of denominator in the interval, the integral function would again be piecewise defined. So, we must be extra careful with the integration area when integrating the tangent.

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