Integration techniques: Integration by Substitution

Integration is the reverse of differentiation, but is quite a lot more frustrating for most people. This is the second of a series of articles on integration techniques.

Integration by Substitution is used when you have a nested functions where one is the derivative of the other. For example, ∫3x²sin(x³) dx  or ∫2x(x²+1)³ dx.

∫g'(x) f(g(x)) dx

Step 1: See if your integral is a composite function

Is your integral a composite functions, such as

f(g(x)), where the derivative of the inner function

g(x) also appears? This is often a function nested within another function, as in the sample general form above. These integrals often contain functions like polynomials, trigonometric functions, or exponentials.

If your integral is the product of two functions rather than nested functions, you will want to use integration by parts instead of integration by substitution.

Step 2: Choose a Substitution, u

u will be the innermost bit of the nested function. For example, in the function

∫3x²sin(x³) dx

u = x³

would be a good substitution because the derivative of x³ is 3x².

Step 3: Calculate du

If u = x³, then taking the derivative of both sides of the equal sign gives us
 du = 3x² dx

It is important after taking the derivative of u that the dx side of the equation for du equals everything outside of f(g(x)). You may have to do a bit of algebra to get them to match.

Alternately, solve so dx is alone on its side and substitute for only the dx.

Step 4: Substitute u and du into your integral

Plugging u and du into the integral should give a much easier integral to solve. If it doesn’t, try choosing something else for u or make sure you are using the appropriate techniques for the integral. 

∫3x²sin(x³) dx

can be rearranged as

∫sin(x³) 3x² dx

which makes it easier to see that the substitution gives us:

∫sin(u) du

which is much easier to solve!

Step 5: Solve the New Integral

We now have a new, easier integral to solve. 

∫sin(u) du = −cos(u) + C

Step 5: Substitute back from u to x

We now just need to substitute back to get our answer in terms of the variable that we started with:

∫sin(u) du = −cos(x³) + C

At this point, you are either finished or you will need to simplify your answer. In our example, we are finished.

Learning when to use which technique takes time, and struggling a bit at first is normal. Allow plenty of time so you don’t stress yourself with a time crunch, and find some practice problems to do. Some good sources are the odd-numbered exercises in the back of each chapter of your textbook, because the answers to the odds are usually in the back of the book, or a book of practice problems like Schaum’s Outline for Calculus or the Calculus For Dummies Workbook.

I will be adding another example or two of Integration by Substitution. If you have questions about this or think of anything you would like to see covered, please leave a comment below!