The fundamental purpose of substitution is to delete the original root sign from the formula.

For example:

The integrand function contains the radical √ (a 2-x 2), so x = asint and the source is a * cost.

The key to simplify indefinite integral with the second kind of method of substitution is to choose a suitable transformation formula x = φ(t). This method is mainly to find the indefinite integral of irrational function (function with root sign). Because it is difficult to integrate the sum root, we try to eliminate the root by substitution to make it easy to calculate.

Let me briefly introduce the methods commonly used in the second alternative method:

(1) radical substitution: the integrand has the radical √(ax+b), which can directly make t = √ (ax+b);

(2) Trigonometric substitution: using trigonometric function substitution, variable root integral is rational function integral, and there are three kinds:

The integrand function contains the radical √ (a 2-x 2), so x = asint.

The integrand function contains the radical √ (a 2+x 2), so x = atant.

The integrand function contains the radical √ (x 2-a 2), so x = asect.

Note: Remember that triangular graphs can facilitate variable recovery.