Solve ∫ (from 1 to 15) of (x^3+1) wrt x

$$\int _ { 1 } ^ { 15 } ( x ^ { 3 } + 1 ) d x$$

Evaluate

$$12670$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{3}+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{3}\mathrm{d}x+\int 1\mathrm{d}x$$

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}.

$$\frac{x^{4}}{4}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{4}}{4}+x$$

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.

$$\frac{15^{4}}{4}+15-\left(\frac{1^{4}}{4}+1\right)$$

Simplify.

$$12670$$