Solve ∫ (from 0 to 100) of (3x^2+y-6x) wrt x

$$\int _ { 0 } ^ { 100 } ( 3 x ^ { 2 } + y - 6 x ) d x$$

Evaluate

$$100\left(y+9700\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3x^{2}+y-6x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{2}\mathrm{d}x+\int y\mathrm{d}x+\int -6x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{2}\mathrm{d}x+\int y\mathrm{d}x-6\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 3 times \frac{x^{3}}{3}.

$$x^{3}+\int y\mathrm{d}x-6\int x\mathrm{d}x$$

Find the integral of y using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{3}+yx-6\int x\mathrm{d}x$$

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 -6 times \frac{x^{2}}{2}.

$$x^{3}+yx-3x^{2}$$

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.

$$100^{3}+y\times 100-3\times 100^{2}-\left(0^{3}+y\times 0-3\times 0^{2}\right)$$

Simplify.

$$970000+100y$$

Differentiate w.r.t. y

$$100$$