Solve ∫ (from 0 to 4) of (10+6x^2) wrt x

$$\int _ { 0 } ^ { 4 } ( 10 + 6 x ^ { 2 } ) d x$$

Evaluate

$$168$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 10+6x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 10\mathrm{d}x+\int 6x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 10\mathrm{d}x+6\int x^{2}\mathrm{d}x$$

Find the integral of 10 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$10x+6\int x^{2}\mathrm{d}x$$

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

$$10x+2x^{3}$$

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.

$$10\times 4+2\times 4^{3}-\left(10\times 0+2\times 0^{3}\right)$$

Simplify.

$$168$$