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1

Solved example of indefinite integrals

$\int x\left(x^2-3\right)dx$
2

We can solve the integral $\int x\left(x^2-3\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x^2-3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=x^2-3$

Find the derivative

$\frac{d}{dx}\left(x^2-3\right)$

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-3\right)$

The derivative of the constant function ($-3$) is equal to zero

$\frac{d}{dx}\left(x^2\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2x$
3

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

$du=2xdx$
4

Isolate $dx$ in the previous equation

$\frac{du}{2x}=dx$

Simplify the fraction $\frac{xu}{2x}$ by $x$

$\int\frac{u}{2}du$
5

Substituting $u$ and $dx$ in the integral and simplify

$\int\frac{u}{2}du$
6

Take the constant $\frac{1}{2}$ out of the integral

$\frac{1}{2}\int udu$
7

Divide $1$ by $2$

$\frac{1}{2}\int udu$

$\frac{1}{2}\cdot \frac{1}{2}u^2$

Multiply $\frac{1}{2}$ times $\frac{1}{2}$

$\frac{1}{4}u^2$
8

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

$\frac{1}{4}u^2$

$\frac{1}{4}\left(x^2-3\right)^2$
9

Replace $u$ with the value that we assigned to it in the beginning: $x^2-3$

$\frac{1}{4}\left(x^2-3\right)^2$
10

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{4}\left(x^2-3\right)^2+C_0$

Final Answer

$\frac{1}{4}\left(x^2-3\right)^2+C_0$

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