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# Indefinite Integrals Calculator

## Get detailed solutions to your math problems with our Indefinite Integrals step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

Here, we show you a step-by-step solved example of indefinite integrals. This solution was automatically generated by our smart calculator:

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

$u=x^2-3$

Differentiate both sides of the equation $u=x^2-3$

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

Find the derivative

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

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

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

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

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 $n=1$

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

When multiplying two powers that have the same base ($\frac{1}{2}$), you can add the exponents

$\left(\frac{1}{2}\right)^2u^2$

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

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

Simplify the expression

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

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

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

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

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