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\int 4x+5\mathrm{d}x
Evaluate the indefinite integral first.
\int 4x\mathrm{d}x+\int 5\mathrm{d}x
Integrate the sum term by term.
4\int x\mathrm{d}x+\int 5\mathrm{d}x
Factor out the constant in each of the terms.
2x^{2}+\int 5\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 4 times \frac{x^{2}}{2}.
2x^{2}+5x
Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax.
2\times 4^{2}+5\times 4-\left(2\times 1^{2}+5\times 1\right)
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.
45
Simplify.