When a function’s argument (that’s the function’s input) is more complicated than something like 3x + 2 (a linear function of x — that is, a function where x is raised to the first power), you can use the substitution method. This method works when the integrand contains a function and the derivative of the function’s argument — in other words, when it contains that extra thing produced by the chain rule — or something just like it except for a constant. And the integrand must not contain anything else. (If that sounds like gibberish, it’ll become clear when you read the following example).
Find the derivative of
![image0.png](https://www.dummies.com/wp-content/uploads/204663.image0.png)
with the substitution method.
Set u equal to the argument of the main function.
Take the derivative of u with respect to x.
Solve for dx.
Make the substitutions.
Antidifferentiate by using the simple reverse rule.
Substitute x-squared back in for u — coming full circle.
If the original problem had been
![image7.png](https://www.dummies.com/wp-content/uploads/204670.image7.png)
Now, you finish this problem just as you did in the preceding Steps 5 and 6, except for the extra 5/2.
![image8.png](https://www.dummies.com/wp-content/uploads/204671.image8.png)
Because C is any old constant,
![image9.png](https://www.dummies.com/wp-content/uploads/204672.image9.png)
You should check this by differentiating it.