In differential calculus, we have learned about the derivative of a function, which is essentially the slope of the tangent of the function at any given point. Like most concepts in math, there is also an opposite, or an inverse. An integral is the inverse of a derivative. Graphically, an integral describes the area underneath a curve on a two-axis graph, and it has a set of unique properties of its own.
While the derivative of a function is represented by
\(\frac{d}{dx}\)
or an apostrophe (
\('\)
), an integral is represented by the
\(\int\)
symbol. Since integrals are the opposite of derivatives, for certain functions, we can use derivative tables in reverse to convert them. For example, we know that the derivative of
\(\sin{x}\)
is
\(\cos{x}\)
. This means that the integral of
\(\cos{x}\)
is
\(\sin{x}\)
.
Examples of Integration by Substitution
One of the most important rules for finding the integral of a functions is integration by substitution, also called U-substitution. In fact, this is the inverse of the chain rule in differential calculus. To use integration by substitution, we need a function that follows, or can be transformed to, this specific form:
\(\int f(g(x))g'(x) \, dx\)
. The most important thing to note here is that we need both
\(g(x)\)
and its derivative
\(g'(x)\)
appearing in the function.
Let’s consider an example:
\(\int \sin{({x}^{2})}(2x) \, dx\)
. Does this match the form we need for integration by substitution? Yes, because the derivative of
\(\sin{x}\)
is
\(\cos{x}\)
, and it immediately follows
\(\sin{x}\)
in the equation.
Next, we identify what
\(f\)
and
\(g\)
are. We can let
\(f(g(x))=\sin{(g(x))}\)
and
\(g(x)={x}^{2}\)
. This means we can use U-substitution, which allows us to substitute
Now, we integrate the function. After integration, we no longer need the
\(\int\)
and the
\(du\)
. This yields
\(-\cos{u}+C\)
. Putting
\(u={x}^{2}\)
back into the equation gives the final answer:
\(-\cos{({x}^{2})}+C\)
But where did this "
\(C\)
" come from? The answer is the following. We know that the derivative of any constant is zero. It allows us to write the derivative of
\({x}^{2}\)
,
\({x}^{2}+1\)
, or
\({x}^{2}+50\)
as simply
\(2x\)
. However, when we reverse this process (integration), we would have no way of knowing what constant we started with. As a result, we add
\(+C\)
to account for this unknown constant.
More Practice
Want to get better at computing integrals and using U-substitution? Try our practice problems above. You can get step-by-step help and see which derivative and integral rules apply to a given function, then try to solve other problems on your own.
Once you are confident about using integration by substitution, you can try tackling other online practice problems, or try the Cymath homework helper app for iOS and Android for explanations and assistance anytime, anywhere.
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