Solve ∫ (from 0 to 5) of 56x^2-x^4

$$\int _ { 0 } ^ { 5 } 56 x ^ { 2 } - x ^ { 4 }$$

Evaluate

$$\frac{5125}{3}\approx 1708.333333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 56x^{2}-x^{4}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 56x^{2}\mathrm{d}x+\int -x^{4}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$56\int x^{2}\mathrm{d}x-\int x^{4}\mathrm{d}x$$

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

$$\frac{56x^{3}}{3}-\int x^{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply -1 times \frac{x^{5}}{5}.

$$\frac{56x^{3}}{3}-\frac{x^{5}}{5}$$

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.

$$\frac{56}{3}\times 5^{3}-\frac{5^{5}}{5}-\left(\frac{56}{3}\times 0^{3}-\frac{0^{5}}{5}\right)$$

Simplify.

$$\frac{5125}{3}$$