Solve ∫ (from -5 to 5) of 9x^3-x^4 wrt x

$$\int_{ -5 }^{ 5 } 9 { x }^{ 3 } - { x }^{ 4 } d x$$

Evaluate

$$-1250$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

$$\int 9x^{3}\mathrm{d}x+\int -x^{4}\mathrm{d}x$$

Factor out the constant in each of the terms.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 9 times \frac{x^{4}}{4}.

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

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}.

$$\frac{9x^{4}}{4}-\frac{x^{5}}{5}$$

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.

$$\frac{9}{4}\times 5^{4}-\frac{5^{5}}{5}-\left(\frac{9}{4}\left(-5\right)^{4}-\frac{\left(-5\right)^{5}}{5}\right)$$

Simplify.

$$-1250$$