Solve ∫ (from 1 to 5) of (2x^5-3x^2)

$$\int _ { 1 } ^ { 5 } ( 2 x ^ { 5 } - 3 x ^ { 2 } )$$

Evaluate

$$5084$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}. Multiply 2 times \frac{x^{6}}{6}.

$$\frac{x^{6}}{3}-3\int x^{2}\mathrm{d}x$$

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}.

$$\frac{x^{6}}{3}-x^{3}$$

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.

$$\frac{5^{6}}{3}-5^{3}-\left(\frac{1^{6}}{3}-1^{3}\right)$$

Simplify.

$$5084$$