Solve ∫ (from 0 to L) of (-10x^2-5/9x^4) wrt x

$$\int _ { 0 } ^ { L } ( - 10 x ^ { 2 } - \frac { 5 } { 9 } x ^ { 4 } ) d x$$

Evaluate

$$-\frac{L^{5}}{9}-\frac{10L^{3}}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -10x^{2}-\frac{5x^{4}}{9}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -10x^{2}\mathrm{d}x+\int -\frac{5x^{4}}{9}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-10\int x^{2}\mathrm{d}x-\frac{5\int x^{4}\mathrm{d}x}{9}$$

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 -10 times \frac{x^{3}}{3}.

$$-\frac{10x^{3}}{3}-\frac{5\int x^{4}\mathrm{d}x}{9}$$

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 -\frac{5}{9} times \frac{x^{5}}{5}.

$$-\frac{10x^{3}}{3}-\frac{x^{5}}{9}$$

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.

$$-\frac{10}{3}L^{3}-\frac{L^{5}}{9}-\left(-\frac{10}{3}\times 0^{3}-\frac{0^{5}}{9}\right)$$

Simplify.

$$-\frac{10L^{3}}{3}-\frac{L^{5}}{9}$$

Differentiate w.r.t. L

$$-\frac{5L^{4}}{9}-10L^{2}$$