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

$$\int _ { 0 } ^ { a } ( a ^ { 3 } - x ^ { 3 } ) d x$$

Evaluate

$$\frac{3a^{4}}{4}$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply -1 times \frac{x^{4}}{4}.

$$a^{3}x-\frac{x^{4}}{4}$$

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.

$$a^{3}a-\frac{a^{4}}{4}-\left(a^{3}\times 0-\frac{0^{4}}{4}\right)$$

Simplify.

$$\frac{3a^{4}}{4}$$

Differentiate w.r.t. a

$$3a^{3}$$