Solve ∫ (from 1 to 3) of (3x^3-2x+5) wrt x

$$\int_{ 1 }^{ 3 } (3 { x }^{ 3 } -2x+5) d x$$

Evaluate

$$62$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 3 times \frac{x^{4}}{4}.

$$\frac{3x^{4}}{4}-2\int x\mathrm{d}x+\int 5\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply -2 times \frac{x^{2}}{2}.

$$\frac{3x^{4}}{4}-x^{2}+\int 5\mathrm{d}x$$

Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{3x^{4}}{4}-x^{2}+5x$$

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.

$$\frac{3}{4}\times 3^{4}-3^{2}+5\times 3-\left(\frac{3}{4}\times 1^{4}-1^{2}+5\times 1\right)$$

Simplify.

$$62$$