Integral: Integration by Substitution

\(\int (\sin{({x}^{2})})x \, dx\)
\(\int \cos{(\cos{2x})}\sin{2x} \, dx\)
\(\int \frac{\sin^{3}x}{\cos{x}} \, dx\)
\(\int {7}^{2x+3} \, dx\)
\(\int \tan^{2}(3x) \, dx\)
\(\int 2y\times \frac{x}{{x}^{2}+3} \, dx\)
\(\int \frac{4x+8}{2{x}^{2}+8x+3} \, dx\)
\(\int {(3-x)}^{10} \, dx\)
\(\int \sqrt{7x+9} \, dx\)

Integration By Substitution - Introduction

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
or an apostrophe (
), an integral is represented by the
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
. This means that the integral of

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
and its derivative
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
, and it immediately follows
in the equation.
Next, we identify what
are. We can let
. This means we can use U-substitution, which allows us to substitute
\(2x \, dx\)
The result?
\(\int \sin{({x}^{2})}(2x) \, dx=\int \sin{u} \, du\)
Now, we integrate the function. After integration, we no longer need the
and the
. This yields
. Putting
back into the equation gives the final answer:
But where did this "
" 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
, or
as simply
. However, when we reverse this process (integration), we would have no way of knowing what constant we started with. As a result, we add
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.