Solve ∫ (from 0 to 3) of 124x^5-x^2+3

$$\int _ { 0 } ^ { 3 } 124 x ^ { 5 } - x ^ { 2 } + 3$$

Evaluate

$$15066$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 124x^{5}-x^{2}+3\mathrm{d}x$$

Integrate the sum term by term.

$$\int 124x^{5}\mathrm{d}x+\int -x^{2}\mathrm{d}x+\int 3\mathrm{d}x$$

Factor out the constant in each of the terms.

$$124\int x^{5}\mathrm{d}x-\int x^{2}\mathrm{d}x+\int 3\mathrm{d}x$$

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}. Multiply 124 times \frac{x^{6}}{6}.

$$\frac{62x^{6}}{3}-\int x^{2}\mathrm{d}x+\int 3\mathrm{d}x$$

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 -1 times \frac{x^{3}}{3}.

$$\frac{62x^{6}}{3}-\frac{x^{3}}{3}+\int 3\mathrm{d}x$$

Find the integral of 3 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{62x^{6}}{3}-\frac{x^{3}}{3}+3x$$

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.

$$\frac{62}{3}\times 3^{6}-\frac{3^{3}}{3}+3\times 3-\left(\frac{62}{3}\times 0^{6}-\frac{0^{3}}{3}+3\times 0\right)$$

Simplify.

$$15066$$