We can solve the integral $\int\left(3x+5\right)\left(2x+3\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
The integral of a constant is equal to the constant times the integral's variable
$\int2xdx+3x$
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The integral of a function times a constant ($2$) is equal to the constant times the integral of the function
$2\int xdx+3x$
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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 $n=1$
$1x^2+3x$
Intermediate steps
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Now replace the values of $u$, $du$ and $v$ in the last formula
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$2x^{3}+\frac{1}{2}x^2+3x\left(3x+5\right)+C_0$
Final answer to the problem
$2x^{3}+\frac{1}{2}x^2+3x\left(3x+5\right)+C_0$
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Integration assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.