Solve ∫ (from 3 to 5) of 3x^2-x^5

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

Evaluate

$$-\frac{7154}{3}\approx -2384.666666667$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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 -1 times \frac{x^{6}}{6}.

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

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.

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

Simplify.

$$-\frac{7154}{3}$$