Here, we show you a step-by-step solved example of indefinite integrals. This solution was automatically generated by our smart calculator:
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
Differentiate both sides of the equation $u=x^2-3$
Find the derivative
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of the constant function ($-3$) is equal to zero
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
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 finding the derivative of the equation above
Isolate $dx$ in the previous equation
Simplify the fraction $\frac{xu}{2x}$ by $x$
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
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 $n=1$
When multiplying two powers that have the same base ($\frac{1}{2}$), you can add the exponents
The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$
Simplify the expression
Replace $u$ with the value that we assigned to it in the beginning: $x^2-3$
Replace $u$ with the value that we assigned to it in the beginning: $x^2-3$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Access detailed step by step solutions to thousands of problems, growing every day!
Most popular problems solved with this calculator: