Solved example of integrals of polynomial functions
Expand the integral $\int\left(x^2+2x+1\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$
The integral $\int x^2dx$ results in: $\frac{x^{3}}{3}$
The integral of a function times a constant ($2$) is equal to the constant times the integral of the function
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 $n=1$
Any expression multiplied by $1$ is equal to itself
The integral $\int2xdx$ results in: $x^2$
The integral of a constant is equal to the constant times the integral's variable
The integral $\int1dx$ results in: $x$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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