Solve ∫ (from - 3 to 2) of x^2-9^2y

$$\int _ { - 3 } ^ { 2 } x ^ { 2 } - 9 ^ { 2 } y$$

Evaluate

$$\frac{35}{3}-405y$$

Solution Steps

Calculate 9 to the power of 2 and get 81.

$$\int _{-3}^{2}x^{2}-81y\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int x^{2}-81y\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int -81y\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x-81\int y\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}-81\int y\mathrm{d}x$$

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

$$\frac{x^{3}}{3}-81yx$$

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.

$$\frac{2^{3}}{3}-81y\times 2-\left(\frac{\left(-3\right)^{3}}{3}-81y\left(-3\right)\right)$$

Simplify.

$$\frac{35}{3}-405y$$

Differentiate w.r.t. y

$$-405$$