Evaluate
-2700
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\int 45-30x\mathrm{d}x
Evaluate the indefinite integral first.
\int 45\mathrm{d}x+\int -30x\mathrm{d}x
Integrate the sum term by term.
\int 45\mathrm{d}x-30\int x\mathrm{d}x
Factor out the constant in each of the terms.
45x-30\int x\mathrm{d}x
Find the integral of 45 using the table of common integrals rule \int a\mathrm{d}x=ax.
45x-15x^{2}
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 -30 times \frac{x^{2}}{2}.
45\times 15-15\times 15^{2}-\left(45\times 0-15\times 0^{2}\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.
-2700
Simplify.
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