Integration by u-Substitution

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Integration by u-Substitution

Up until now, we have only been able to integrate relatively straightforward functions. What if we had something a bit more complicated? One way we can try to integrate is by u-substitution. Let’s look at an example:

Example 1:

Evaluate the integral:

Something to notice about this integral is that it consists of both a function f (x2+5) and the derivative of that function, f ‘ (2x). This can be a but unwieldy to integrate, so we can substitute a variable in. We will take whichever function is f and let it equal u.

If you are having trouble determining which one is which, as a general rule we want u to be the more complicated function, such as the inner part of a composite function, or a function raised to a power.

In this case, we will let u = x2+5:

Now we have an integral with two separate variables, u and x. To remedy this, we need to take the derivative of u, solve it for dx, and substitute it in. Since we are taking the derivative of u, after taking the derivative of any x we need to put the term dx:

Next, we can plug in the substitution for dx:

Now, the 2x can be canceled out, leaving only a function in terms of u:

Sometimes, we might run into a problem here. If, at this point, there are still x terms, or we get an equation that cannot be integrated, then there are two possibilities (aside from a mistake made when working the problem): either the wrong term was chosen for u, or the integral cannot be solved with u-substitution. This one looks like it works out, however.

Now we can integrate:

There is one more step; u was just a substitute variable, so we need to plug in the original function which u represented:

Example 2:

Evaluate the integral:

First, we need to pick a u and find du; we will let u = x3-7:

Now we can begin our substitutions, starting with u:

It is important to note that it doesn’t matter if the constants don’t cancel when we put in, since they can be factored out; what is important is that all the x’s get canceled.

Now we can integrate:

Now, we just need to put the u term back in:

Example 3:

Evaluate the integral:

First, we need to find u, du, and dx. Let’s let u = 3x-2:

Now, we can make our substitutions:

Now that we have the integral of a function with respect to u, we can integrate:

All that we need to do now in substitute out the u:

Example 4:

Evaluate the integral:

For this example, we can let u equal sin x or cos x. Let’s start by letting it equal sin x:

Now, we can make our substitutions:

Next, we can integrate and substitute out the u:

Now, let’s try the same function, this time letting u = cos x:

Next, substituting and integrating:

Notice that the answer changes depending on whether we let u equal sin x or cos x. If we plug in a value for the x of each answer, we find that they are separated by exactly one half. This is accounted for in the C that we add; if we differentiate either answer, they both still come out to the original equation.

Example 5:

Evaluate the definite integral:

Let’s start by letting u = 4×2-4:

Now we can substitute:

Next, we can integrate and substitute out u:

Now, to solve the definite integral, we need to subtract the bottom number from the top number:

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