Solve ∫ (from 1 to 8) of 5x^4 wrt x

$$\int _ { 1 } ^ { 8 } 5 x ^ { 4 } d x$$

Evaluate

$$32767$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5x^{4}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$5\int x^{4}\mathrm{d}x$$

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

$$x^{5}$$

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.

$$8^{5}-1^{5}$$

Simplify.

$$32767$$