Solve ∫ (from - 3 to 1) of 6x^2-5x+2d

$$\int _ { - 3 } ^ { 1 } 6 x ^ { 2 } - 5 x + 2 d$$

Evaluate

$$8d+76$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 6x^{2}-5x+2d\mathrm{d}x$$

Integrate the sum term by term.

$$\int 6x^{2}\mathrm{d}x+\int -5x\mathrm{d}x+\int 2d\mathrm{d}x$$

Factor out the constant in each of the terms.

$$6\int x^{2}\mathrm{d}x-5\int x\mathrm{d}x+2\int d\mathrm{d}x$$

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 6 times \frac{x^{3}}{3}.

$$2x^{3}-5\int x\mathrm{d}x+2\int d\mathrm{d}x$$

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 -5 times \frac{x^{2}}{2}.

$$2x^{3}-\frac{5x^{2}}{2}+2\int d\mathrm{d}x$$

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

$$2x^{3}-\frac{5x^{2}}{2}+2dx$$

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.

$$2\times 1^{3}-\frac{5}{2}\times 1^{2}+2d\times 1-\left(2\left(-3\right)^{3}-\frac{5}{2}\left(-3\right)^{2}+2d\left(-3\right)\right)$$

Simplify.

$$76+8d$$

Differentiate w.r.t. d

$$8$$