Integration of combined functions

Many integration rules are obtained directly from derivative rules. When we differentiate combined functions, it is important to identify what is an “inner function” and what is an “outer function”. For example, in a function

The square root is an outer function and 1 + 2x is an inner function. The function f(x) can also be written

This needs to be well understood before one can go ahead. The function f(x) is called a “combined function” and is differentiated by a “chain rule”

This gives the so-called combined function integration rule

The same thing is often written in a different form

[You can always check the integration rules by taking the derivative from the right side of the equation, the result should be a function inside the integral.]

This rule seems difficult, but it is better understood with examples and exercises. The most important thing to remember is that the integral can only be calculated when the derivative of the internal function h '(x) is also found in the integral.

Example 1: Integrate

Now inside the integral there is a derivative of the inner function of the square root expression 2x and we can use the integral rule. We no longer have to find the integral function of the square root. This is again done by the integration rule of the fractional power and the power function.

It is advisable to check this and other corresponding integral calculations by deriving the fractional power form obtained (second to last line of this example).

Keep the integral rule of the combined function in mind in the following chapters when we are integrating exponential functions and trigonometric functions.

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