Solve ∫ (from 1 to 4) of x^5+x^9

$$\int _ { 1 } ^ { 4 } x ^ { 5 } + x ^ { 9 }$$

Evaluate

$$105540$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{5}+x^{9}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{5}\mathrm{d}x+\int x^{9}\mathrm{d}x$$

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}.

$$\frac{x^{6}}{6}+\int x^{9}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{9}\mathrm{d}x with \frac{x^{10}}{10}.

$$\frac{x^{6}}{6}+\frac{x^{10}}{10}$$

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.

$$\frac{4^{6}}{6}+\frac{4^{10}}{10}-\left(\frac{1^{6}}{6}+\frac{1^{10}}{10}\right)$$

Simplify.

$$105540$$