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1

Solved example of integral

$\int 3x^2+5x+2dx$
2

Expand the integral $\int\left(3x^2+5x+2\right)dx$

$\int3x^2dx+\int5xdx+\int2dx$

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$3\int x^2dx$

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 $2$

$x^{3}$
3

The integral $\int3x^2dx$ results in: $x^{3}$

$x^{3}$

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$5\int xdx$

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 $n=1$

$\frac{5}{2}x^2$
4

The integral $\int5xdx$ results in: $\frac{5}{2}x^2$

$\frac{5}{2}x^2$

The integral of a constant is equal to the constant times the integral's variable

$2x$
5

The integral $\int2dx$ results in: $2x$

$2x$
6

Gather the results of all integrals

$x^{3}+\frac{5}{2}x^2+2x$
7

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

$x^{3}+\frac{5}{2}x^2+2x+C_0$

Final Answer

$x^{3}+\frac{5}{2}x^2+2x+C_0$

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