Solve ∫ (from 4 to 8) of 4x^3-5

$$\int _ { 4 } ^ { 8 } 4 x ^ { 3 } - 5$$

Evaluate

$$3820$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4x^{3}-5\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4x^{3}\mathrm{d}x+\int -5\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\int x^{3}\mathrm{d}x+\int -5\mathrm{d}x$$

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 4 times \frac{x^{4}}{4}.

$$x^{4}+\int -5\mathrm{d}x$$

Find the integral of -5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{4}-5x$$

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.

$$8^{4}-5\times 8-\left(4^{4}-5\times 4\right)$$

Simplify.

$$3820$$