Substitution

Substitution is the most common integral technique in elementary calculus. Recall what we mentioned in common integrals that we use $u$ and $\mathrm{d}u$ instead of $x$ and $\mathrm{d}x$, and this is because $u$ may represent an entire function, not just a single variable. We will tackle with this more specifically in this section.

The Substitution Rule

We will start with this integral question:

$$\int{3x^2\sin{x^3}}\ \mathrm{d}x=\ ?$$

Note that there is an product, which is beyond all common integrals, so we need to simplify to an easier form. An example is below:

Solution

Let: $u=x^3$

$$\begin{align*} \frac{\mathrm{d}u}{\mathrm{d}x}&=3x^2\\ \color{red}{\mathrm{d}u}&=\color{red}{3x^2\ \mathrm{d}x}\\ \end{align*}$$

Plug in back to the question:

$$\begin{align*} \int{\color{red}{3x^2}}\sin{x^3}\ \color{red}{\mathrm{d}x}&=\int{\sin{u}}\ \color{red}{\mathrm{d}u}\\ &=\int{\sin{u}}\ \mathrm{d}u\\ &=-\cos{u}+C\\ &=-\cos{x^3}+C \end{align*}$$

Hence:

$$\int{3x^2\sin{x^3}}\ \mathrm{d}x=-\cos{x^3}+C \tag*{$\blacksquare$}$$

By replacing the variable $x$ into a function of $u$, we can simplify the overall question to a common integral. As you can see, this is a method we inversely use the Chain Rule, and generally, it is called:

The substitution rule

Given a differenciable function $u=g(x)$, and $f$ is continuous on its interval, then:

$$\int{f\left(g\left(x\right)\right)g'(x)}\ \mathrm{d}x=\int{f\left(u\right)}\ \mathrm{d}u$$

Definite Integral

In the case of Definite Integral, the bound should also changes, as:

$$\int_{a}^{b}{f\left(g\left(x\right)\right)g'(x)}\ \mathrm{d}x=\int_{g(a)}^{g(b)}{f\left(u\right)}\ \mathrm{d}u$$

Direct Substitution

Direct Substitution is a method created by the author himself. It simplify the process of substitution without write out $u$ and $\mathrm{d}u$. I will show this with an example:

$$\int{3x^2\sin{x^3}}\ \mathrm{d}x=\ ?$$

This time, let's try what I called Direct Substituton:

Direct Substitution Approach

After analyze, we want to substitute $x^3$. All we need to do is substitude $\mathrm{d}x$ to a fraction where the top is $\mathrm{d}x^3$ and the bottom is the derivative of $x^3$ based on $x$, as:

$$\int{3x^2\sin{x^3}}\ \mathrm{d}x=\int{\cancel{3x^2}\sin{x^3}}\ \frac{\mathrm{d}(x^3)}{\cancel{3x^2}}=\int{\sin{x^3}}\ \mathrm{d}(x^3)$$

And all we need is to consider $x^3$ as $u$ and integrate normally:

$$\implies \int{\sin{x^3}}\ \mathrm{d}(x^3)=-\cos{x^3}+C$$

More importantly, for definite integral, it doesn't need to change bounds, as:

$$\int_{a}^{b}3x^2\sin{x^3}\ \mathrm{d}x=\int_{a}^{b}{\sin{x^3}}\ \mathrm{d}(x^3)=-\cos{x^3}\big|_{a}^{b}=-\cos{b^3}+\cos{a^3}$$

Direct Substitution

In general, if we want to substitude $ax+b$ or term that can cancel the other one, we may use direct substitution

$$\int{{f(x)}\underbrace{g(x)}_{\text{Substitude}}}\ \mathrm{d}x=\int{f(x)g(x)}\ \frac{\mathrm{d}(g(x))}{g'(x)}$$

Back Substitution

For some integrals, after we made the $u$ substitution, we might need to express $x$ as a function of $u$. See more example in Integral with √(ax+b).

Trigonometry Substitution