Solve ∫ (from 0 to a) of -3x^2+3 wrt x

$$\int _ { 0 } ^ { a } - 3 x ^ { 2 } + 3 d x$$

Evaluate

$$a\left(3-a^{2}\right)$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

Find the integral of 3 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

$$-a^{3}+3a-\left(-0^{3}+3\times 0\right)$$

Simplify.

$$a\left(-a^{2}+3\right)$$

Differentiate w.r.t. a

$$3-3a^{2}$$