Final answer to the problem
$\frac{x^{3}}{3}-\frac{3}{2}x^2-4x+C_0$
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Step-by-step Solution
Specify the solving method
1
Find the integral
$\int\left(x+1\right)\left(x-4\right)dx$
2
Rewrite the integrand $\left(x+1\right)\left(x-4\right)$ in expanded form
$\int\left(x^2-3x-4\right)dx$
3
Expand the integral $\int\left(x^2-3x-4\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
$\int x^2dx+\int-3xdx+\int-4dx$
4
The integral $\int x^2dx$ results in: $\frac{x^{3}}{3}$
$\frac{x^{3}}{3}$
5
The integral $\int-3xdx$ results in: $-\frac{3}{2}x^2$
$-\frac{3}{2}x^2$
6
The integral $\int-4dx$ results in: $-4x$
$-4x$
7
Gather the results of all integrals
$\frac{x^{3}}{3}-\frac{3}{2}x^2-4x$
8
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{x^{3}}{3}-\frac{3}{2}x^2-4x+C_0$
Final answer to the problem
$\frac{x^{3}}{3}-\frac{3}{2}x^2-4x+C_0$